# An infinite question: Why doesn't  .999~ = 1?

**URL:** <https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711>\
**Category:** Cecil's Columns/Staff Reports\
**Created:** [September 24, 2016, 7:29am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711 "2016-09-24T07:29:45Z")\
**Posts on this page:** 20\
**Page:** 18

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**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\
**Post date:** [October 30, 2016, 5:27pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/341 "2016-10-30T17:27:17Z")

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It’s not a process, it’s a number … there’s a difference …

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**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [October 30, 2016, 6:07pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/342 "2016-10-30T18:07:06Z")

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> [@netzweltler](#):
>
> That’s ridiculous. Apply the process suggested and you will see what really happens.
> 
> t = 0: I move my pen from point 0 to point 0.25 of the number line  
> t = 0.9: I move my pen from point 0.25 to point 0.250 of the number line  
> t = 0.99: I move my pen from point 0.250 to point 0.2500 of the number line  
> …
> 
> I have completed the task adding infinitely many zeros by t = 1. The position of the pen hasn’t changed and is equal to 1/4 from t = 0 to t = 1.
> 
> So, “by the reasoning presented” 1/4 **is** equal to 0.25000~.

**You’ve previously insisted that the sequence on the left (0, 0.9, 0.99, … ) NEVER reaches 1.**  
I’m taking it as a victory that you have finally seen the folly of your ways.

Hurrah for persuasion! Maths is once more saved and the world can continue spinning safely on its axis.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [October 31, 2016, 7:46am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/343 "2016-10-31T07:46:03Z")

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> [@The\_Great\_Unwashed](#):
>
> > [@netzweltler](#):
> >
> > That’s ridiculous. Apply the process suggested and you will see what really happens.
> > 
> > t = 0: I move my pen from point 0 to point 0.25 of the number line  
> > t = 0.9: I move my pen from point 0.25 to point 0.250 of the number line  
> > t = 0.99: I move my pen from point 0.250 to point 0.2500 of the number line  
> > …
> > 
> > I have completed the task adding infinitely many zeros by t = 1. The position of the pen hasn’t changed and is equal to 1/4 from t = 0 to t = 1.
> > 
> > So, “by the reasoning presented” 1/4 **is** equal to 0.25000~.
> 
> **You’ve previously insisted that the sequence on the left (0, 0.9, 0.99, … ) NEVER reaches 1.**

Obviously you are talking about the sequence marked in red. t means the time when an action takes place. This doesn’t mean that time doesn’t reach 1, 2, and after. There are just no actions specified for these times.

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<div class="post-metadata">

**Author:** ![watchwolf49](https://avatars.discourse-cdn.com/v4/letter/w/e9c0ed/32.png) [@watchwolf49](https://boards.straightdope.com/u/watchwolf49)\
**Post date:** [October 31, 2016, 12:27pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/344 "2016-10-31T12:27:21Z")

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> [@netzweltler](#):
>
> Obviously you are talking about the sequence marked in red. t means the time when an action takes place. This doesn’t mean that time doesn’t reach 1, 2, and after. There are just no actions specified for these times.

So, you’re just picking and choosing which numbers your process works, and ignoring all the rest … does that sound mathematical to you? I’m pretty sure you’re missing an important step, you still have to integrate the whole shmear (from 0 to ∞). Perhaps **Riemann** would be kind enough to explain …

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 1, 2016, 8:07am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/345 "2016-11-01T08:07:50Z")

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> [@watchwolf49](#):
>
> So, you’re just picking and choosing which numbers your process works, and ignoring all the rest … does that sound mathematical to you? I’m pretty sure you’re missing an important step, you still have to integrate the whole shmear (from 0 to ∞). Perhaps **Riemann** would be kind enough to explain …

I am picking any numbers which make my process work. Any sequence which guarantees that I have completed infinitely many steps by t = 1. It can be t = { 0, 0.9, 0.99, 0.999, … }. It can also be t = { 0, 0.5, 0,75, 0.875, … }.

No step missing.

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<div class="post-metadata">

**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\
**Post date:** [November 1, 2016, 8:37am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/346 "2016-11-01T08:37:49Z")

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> [@netzweltler](#):
>
> I am picking any numbers which make my process work. Any sequence which guarantees that I have completed infinitely many steps by t = 1. It can be t = { 0, 0.9, 0.99, 0.999, … }. It can also be t = { 0, 0.5, 0,75, 0.875, … }.
> 
> No step missing.

Except it obviously doesn’t work, or at least doesn’t work they way you claim it does. If you’ve completed infinitely many steps by the time, **and only by the time,** you reach t=1, then 0.999… = 1

Note the bit in bold. If you have any difference between the “almost infinite” number of nines after the decimal point and 1, then you haven’t reached t=1.

As I said once before, you’re free to decide that infinites are impossible, or that there’s an infinitesimal “last step”, but that is not some deep mathematical truth you have discovered. It’s a choice and a poor one. For instance it means that for your process where the value is 0.9 when t=0.9 and 0.99 when t=0.99 you **cannot** reach 1.

At times it seems like that is your point, you cannot reach t=1, but then you post something like the above, where you say you’ve completed infinitely many steps by t=1. So which is it? And try to give it some thought and examine where we might be misunderstanding you, if anywhere.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 1, 2016, 9:15am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/347 "2016-11-01T09:15:34Z")

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> [@naita](#):
>
> Except it obviously doesn’t work, or at least doesn’t work they way you claim it does. If you’ve completed infinitely many steps by the time, **and only by the time,** you reach t=1, then 0.999… = 1
> 
> Note the bit in bold. If you have any difference between the “almost infinite” number of nines after the decimal point and 1, then you haven’t reached t=1.
> 
> As I said once before, you’re free to decide that infinites are impossible, or that there’s an infinitesimal “last step”, but that is not some deep mathematical truth you have discovered. It’s a choice and a poor one. For instance it means that for your process where the value is 0.9 when t=0.9 and 0.99 when t=0.99 you **cannot** reach 1.
> 
> At times it seems like that is your point, you cannot reach t=1, but then you post something like the above, where you say you’ve completed infinitely many steps by t=1. So which is it? And try to give it some thought and examine where we might be misunderstanding you, if anywhere.

We are reaching t = 1. Time continues. I was stating, we don’t reach point 1 as a part of this process:

t = 0: I move my pen from point 0 to point 0.9 of the number line  
t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line  
t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line  
…

Even when we have reached t = 1 we haven’t reached point 1. Because all the action takes place at t \< 1. There is no action specified for t = 1 and after.

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<div class="post-metadata">

**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\
**Post date:** [November 1, 2016, 9:27am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/348 "2016-11-01T09:27:47Z")

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> [@The\_Great\_Unwashed](#):
>
> [QUOTE=netzweltler]
> 
> That’s why I was asking a few posts ago, if we don’t stop at a specific point on the number line, is there a “point” 0.999… on the number line?

Yes. It’s where 1 is.  
[/QUOTE]

Is this argument still going on? The correct and incorrect answers have each been given multiple times. Doesn’t it (like 0.999999… :rolleyes: ) get _repetitious_?

Let me help! 😛 Let’s [start with this, Zeno’s paradox of Achilles and the tortoise](https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise).

@ **netzweltler** — Do you agree that “In a race, the quickest runner can never overtake the slowest”? If not, tell us what is wrong (or right) with Zeno’s reasoning. When you complete this assignment, connect the dots and relate it to the topic of this thread.

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<div class="post-metadata">

**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\
**Post date:** [November 1, 2016, 10:10am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/349 "2016-11-01T10:10:09Z")

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> [@netzweltler](#):
>
> We are reaching t = 1. Time continues. I was stating, we don’t reach point 1 as a part of this process:
> 
> t = 0: I move my pen from point 0 to point 0.9 of the number line  
> t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line  
> t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line  
> …
> 
> Even when we have reached t = 1 we haven’t reached point 1. Because all the action takes place at t \< 1. There is no action specified for t = 1 and after.

That’s what I thought. So let’s describe a different process, that is similar to yours. I move my pen at a constant speed of one unit on the number line per one unit of time.

Now the “steps” below aren’t steps, they’re just statements of facts about some of the states as we complete the process above.

At t=0 I’m at 0  
At t=0.9 I’m at 0.9  
At t=0.99 I’m at 0.99  
…  
At t=1 I’m at 1.

The starting points of every step in your process has an equivalent state in this process, and in this process we _do_ reach 1, so at some point we passed a point that could be described by infinite nines after the decimal point.

What is your opinion about this process? That we didn’t pass a point that could be described by infinite nines? (What’s the maximum number of nines then?) That that point can’t be defined as anything? (But we did pass it, and we can give a really good description of when.) That that point isn’t 1 because of your step by step process? (But according to you your process never reaches 1, how can it say anything about 1 when it doesn’t reach it?)

Now please don’t answer this following statement unless it’s to accept it’s correct. We already know you disagree with it, so you need to focus on the paragraph above if we’re to stop going in circles. But my, and all of mathematics, claim, is that we only pass that point of infinite nines when t=1. At t=0.9 there is 0.1 time units left until we pass 1, at t=0.99 there is 0.01 time units left, at t=0.999 there is 0.001 time units left. As we approach 0.999…, the one with infinite nines, which must exist by this description of things, we approach 0.000… infinite zeros. With infinite zeros there is no terminating 1, so 0.000… is properly and exactly 0, which means that 0.999… is reached at t=1, which is also when we’re at 1, so 0.999… is equal to 1.

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<div class="post-metadata">

**Author:** ![The\_Great\_Unwashed](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/the_great_unwashed/32/4791_2.png) [@The\_Great\_Unwashed](https://boards.straightdope.com/u/The_Great_Unwashed)\
**Post date:** [November 1, 2016, 10:25am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/350 "2016-11-01T10:25:00Z")

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> [@naita](#):
>
> That’s what I thought. So let’s describe a different process, that is similar to yours. I move my pen at a constant speed of one unit on the number line per one unit of time.

Good luck there, I already tried, he won’t be convinced.

> [@The\_Great\_Unwashed](#):
>
> As far as I can tell, your pen is moving with constant speed. When t is 1 your pen is at 1.

> [@The\_Great\_Unwashed](#):
>
> Usain Bolt runs at about 10 metres (1 decametre) per second (slightly faster, but suppose this is a bad day for him).
> 
> So :  
> at t = 0 he is at 0 decametres  
> at t = 0.9 he is at 0.9 decametres  
> at t = 0.99 he is at 0.99 decametres  
> at t = 0.999 he is at 0.999 decametres  
> at t = 0.9999 he is at 0.9999 decametres
> 
> Somehow you want to conclude that his position at time t = 1 isn’t defined, that he can never cover those ten metres. This is Zeno’s paradox, which has long since been resolved, and which doesn’t surprise anyone but flat-earthers, n00bs and trolls.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 1, 2016, 10:45am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/351 "2016-11-01T10:45:33Z")

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> [@septimus](#):
>
> Is this argument still going on? The correct and incorrect answers have each been given multiple times. Doesn’t it (like 0.999999… :rolleyes: ) get _repetitious_?
> 
> Let me help! 😛 Let’s [start with this, Zeno’s paradox of Achilles and the tortoise](https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise).
> 
> @ **netzweltler** — Do you agree that “In a race, the quickest runner can never overtake the slowest”? If not, tell us what is wrong (or right) with Zeno’s reasoning. When you complete this assignment, connect the dots and relate it to the topic of this thread.

For simplification we should deal with the dichotomy paradox and not the paradox of Achilles and the tortoise in this context.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 1, 2016, 11:02am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/352 "2016-11-01T11:02:34Z")

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> [@naita](#):
>
> That’s what I thought. So let’s describe a different process, that is similar to yours. I move my pen at a constant speed of one unit on the number line per one unit of time.
> 
> Now the “steps” below aren’t steps, they’re just statements of facts about some of the states as we complete the process above.
> 
> At t=0 I’m at 0  
> At t=0.9 I’m at 0.9  
> At t=0.99 I’m at 0.99  
> …  
> At t=1 I’m at 1.
> 
> The starting points of every step in your process has an equivalent state in this process, and in this process we _do_ reach 1, so at some point we passed a point that could be described by infinite nines after the decimal point.
> 
> What is your opinion about this process? That we didn’t pass a point that could be described by infinite nines? (What’s the maximum number of nines then?) That that point can’t be defined as anything? (But we did pass it, and we can give a really good description of when.) That that point isn’t 1 because of your step by step process? (But according to you your process never reaches 1, how can it say anything about 1 when it doesn’t reach it?)
> 
> Now please don’t answer this following statement unless it’s to accept it’s correct. We already know you disagree with it, so you need to focus on the paragraph above if we’re to stop going in circles. But my, and all of mathematics, claim, is that we only pass that point of infinite nines when t=1. At t=0.9 there is 0.1 time units left until we pass 1, at t=0.99 there is 0.01 time units left, at t=0.999 there is 0.001 time units left. As we approach 0.999…, the one with infinite nines, which must exist by this description of things, we approach 0.000… infinite zeros. With infinite zeros there is no terminating 1, so 0.000… is properly and exactly 0, which means that 0.999… is reached at t=1, which is also when we’re at 1, so 0.999… is equal to 1.

As you say, your process is not dealing with steps. So yes, when continuously moving you reach point 1 after passing the infinitely many 9s in 0.999…  
But that’s not what’s going on when you add 0.9 + 0.09 + 0.009 + … We are dealing with steps!

> [@naita](#):
>
> (But according to you your process never reaches 1, how can it say anything about 1 when it doesn’t reach it?)

It can, because my process doesn’t specify an action for t = 1. So, nothing happens at t = 1. Therefore, the state of the pen is solely defined by the actions at t \< 1. At t = { 0, 0.9, 0.99, 0.999, … }.

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<div class="post-metadata">

**Author:** ![Telemark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/telemark/32/372_2.png) [@Telemark](https://boards.straightdope.com/u/Telemark)\
**Post date:** [November 1, 2016, 11:36am UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/353 "2016-11-01T11:36:18Z")

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> [@netzweltler](#):
>
> But that’s not what’s going on when you add 0.9 + 0.09 + 0.009 + … We are dealing with steps!

No. You’re dealing with steps. The rest of the world recognizes that you can’t do that when discussing infinities. It’s pointless and non-productive to do so. You can continue to play this game and get no useful answers or explain why you think your thought experiment is meaningful.

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<div class="post-metadata">

**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\
**Post date:** [November 1, 2016, 12:21pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/354 "2016-11-01T12:21:27Z")

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> [@netzweltler](#):
>
> As you say, your process is not dealing with steps. So yes, when continuously moving you reach point 1 after passing the infinitely many 9s in 0.999…  
> But that’s not what’s going on when you add 0.9 + 0.09 + 0.009 + … We are dealing with steps!

Try reading my description again. All your steps exist as states in my process. Do you agree with that?

And I’m not saying we reach point 1 _after_ passing infinitely many 9s, we reach point 1 _when_ we reach infinitely many 9s.

Do you agree that in the constant speed process the difference between 0.999… and 1 is 0? If not, what is the difference?

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<div class="post-metadata">

**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\
**Post date:** [November 1, 2016, 1:32pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/355 "2016-11-01T13:32:52Z")

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> [@netzweltler](#):
>
> For simplification we should deal with the dichotomy paradox and not the paradox of Achilles and the tortoise in this context.

Wrong. Achilles and the tortoise is the perfect paradox to mimic your thoughts. Answer the questions please.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 1, 2016, 4:51pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/356 "2016-11-01T16:51:06Z")

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> [@Telemark](#):
>
> No. You’re dealing with steps. The rest of the world recognizes that you can’t do that when discussing infinities.

0.999… is dealing with steps. **Infinitely** many of them. And this means discussing **infinity** , of course.

> [@Telemark](#):
>
> It’s pointless and non-productive to do so. You can continue to play this game and get no useful answers or explain why you think your thought experiment is meaningful.

It was meaningful for centuries. Since Zeno came up with that for the first time.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 1, 2016, 4:55pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/357 "2016-11-01T16:55:24Z")

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> [@naita](#):
>
> Try reading my description again. All your steps exist as states in my process. Do you agree with that?

Yes. So what?

> [@naita](#):
>
> And I’m not saying we reach point 1 _after_ passing infinitely many 9s, we reach point 1 _when_ we reach infinitely many 9s.

There is no point when we **reach** infinitely many 9s. What should that be? The **last** point?

> [@naita](#):
>
> Do you agree that in the constant speed process the difference between 0.999… and 1 is 0? If not, what is the difference?

0.999… is not a result of a constant speed process.

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<div class="post-metadata">

**Author:** ![netzweltler](https://avatars.discourse-cdn.com/v4/letter/n/a3d4f5/32.png) [@netzweltler](https://boards.straightdope.com/u/netzweltler)\
**Post date:** [November 1, 2016, 4:56pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/358 "2016-11-01T16:56:39Z")

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> [@septimus](#):
>
> Wrong. Achilles and the tortoise is the perfect paradox to mimic your thoughts. Answer the questions please.

Sorry. Still cannot see how this paradox is related to the problem under discussion. Please explain.

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<div class="post-metadata">

**Author:** ![naita](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/naita/32/5862_2.png) [@naita](https://boards.straightdope.com/u/naita)\
**Post date:** [November 1, 2016, 5:27pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/359 "2016-11-01T17:27:20Z")

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> [@naita](#):
>
> Try reading my description again. All your steps exist as states in my process. Do you agree with that?

> [@netzweltler](#):
>
> Yes. So what?

So the constant speed process encompasses your step by step process.

> [@naita](#):
>
> And I’m not saying we reach point 1 _after_ passing infinitely many 9s, we reach point 1 _when_ we reach infinitely many 9s.

> [@netzweltler](#):
>
> There is no point when we **reach** infinitely many 9s. What should that be? The **last** point?

If you’d paid any intention you’d know that I already consider that point 1. But think about what you’re saying here. Moving from 0 to 1 at a constant speed we pass every possible point on the number line in the interval [0, 1] in time t=[0, 1], this includes 0.9, 0.99, 0.999 and so on. If it doesn’t include infinite 9’s, what’s the largest number of nines it includes?

> [@naita](#):
>
> Do you agree that in the constant speed process the difference between 0.999… and 1 is 0? If not, what is the difference?

> [@netzweltler](#):
>
> 0.999… is not a result of a constant speed process.

0.999… doesn’t have to be the result of **any** process, it’s a number in and of itself. And as I just described logic entails it occurs as a state in the constant speed process. Try to look outside the box you’ve got yourself stuck in here. Think about the process I just described. We’re moving from 0 to 1 on a number line over the span of 1 unit of time. At 0.9 we’re at 0.9, at 0.99 we’re at 0.99 and so on. Unless we can point to a maximum number of nines, during this movement there has to be a point we can describe by an infinite number of nines. Where is that point, and if you think it doesn’t exist, what is the largest number of nines describing a point we do pass by.

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<div class="post-metadata">

**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\
**Post date:** [November 1, 2016, 6:08pm UTC](https://boards.straightdope.com/t/an-infinite-question-why-doesnt-999-1/766711/360 "2016-11-01T18:08:01Z")

</div>

> [@netzweltler](#):
>
> Sorry. Still cannot see how this paradox is related to the problem under discussion. Please explain.

😕_ **The two formulations seem extremely similar:** _

Achilles never reaches the tortoise because his residual distance (1/10, 1/100, 1/1000, …) remains positive after any finite number of steps.

Netzwelter never reaches 1/3 because his residual distance (0.333…, 0.0333, 0.00333…, 0.0003333 …) remains positive after any finite number of states.

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