One is a number. The other is a segment.
I’m not familiar with your notation … maybe I was presented with this material way back when I was a lad … but that was half a century ago … I do know that whatever your trying to say can be said using functions …
… but The Great Unwashed has answered your questions … if you have a point, then make it … frankly, I’m not sure you understand your notation … if I understand this correctly, then I am particularly fond of TGU answer to #3 … [0,1] … that one makes me smile …
Why do you think the value of our previous function cannot equal it’s limit?
ETA: If I may modify BCP answer a bit … 0.9 is a point on the number line …
It’s basic set theoretical notation.
0.999… can be written as [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪…
and there is no segment that includes point 1 in [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪…
You cannot find a segment which is not already present in the list
[0, 0.9]
[0, 0.9]∪[0.9, 0.99]
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]
…
(4) Usually we are saying that a number is greater than a set of smaller numbers, if it contains at least one segment which is not a segment of one of the smaller numbers.
Do you agree to that?
A little exercise for those who are not familiar with set theoretical notation: We agree that 1 is greater than the numbers of the set { 0.5, 0.8, 0.95 }.
In set theoretical notation 1 can be written as [0, 0.5]∪[0.5, 0.8]∪[0.8, 0.95]∪[0.95, 1], the union of these segments. [0.95, 1] is a segment which is not present in one of the numbers { 0.5, 0.8, 0.95 }. The segment [0.95, 1] shows that 1 is greater than the numbers 0.5, 0.8, and 0.95. Now try that with the set { 0.9, 0.99, 0.999, … } an show that 0.999… is greater than all numbers of this set. What is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …?
This is only the case if you reject that a convergent infinite series is equal to its limit. In which case you will find a segment which is not necessarily present in the list: the limit.
So you’ve finally stopped repeating your nursery rhyme, and are speaking mathematics. My impression is you are unfamiliar with open sets and closed sets. And you are correct: 1 (a synonym of 0.9999) is not present in the half-open interval [0,1).
Ho hum. This is all in chapter 1 of any Freshman Topology text. We could have saved a lot of time if you didn’t feel the need to first cut-paste your nursery rhyme 27 times. :smack:
In which case you’re just finding another way in which your mysteriously-moving-pen-non-argument can be written.
Were you hoping no-one was going to notice?
Anyway, speaking of what 0.999… can be written as, it can be written as 1 (didn’t we already mention this?). I think I’ll go with that – it uses less screenspace.
Why do you think the value of our previous function cannot equal it’s limit?
Near as I can figure, you’re just saying the exact same thing here … if you could please explicitly state what the new information is that you are introducing …
Hold on here … excuse my ignorance, but since when can the union of two finite line segments ever be exactly equivalent to a point?
Do you mean the length of the line segment is equal to (0.999…) … then as the number of segments approaches infinity, then the length of the union approaches 1 … which leads us back to my question at the top of this post … the one in bold typeface … in case you didn’t see it there …
(4) - If the set contains points exclusively, then it contains no line segments, so I don’t know what you’re asking me to agree with.
There is no “not necessarily present” in mathematics. Mathematics clearly states that there is no segment that includes point 1 in [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪…
Of course we are talking about the length of segments.
Since you know that now, please try to answer:
In set theoretical notation 1 can be written as [0, 0.5]∪[0.5, 0.8]∪[0.8, 0.95]∪[0.95, 1], the union of these segments. [0.95, 1] is a segment which is not present in one of the numbers { 0.5, 0.8, 0.95 }. The segment [0.95, 1] shows that 1 is greater than the numbers 0.5, 0.8, and 0.95. Now try that with the set { 0.9, 0.99, 0.999, … } an show that 0.999… is greater than all numbers of this set. What is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …?
Why do you think the value of our previous function cannot equal it’s limit?
Why do you think the limit point of the set doesn’t exist in the set?
Aren’t we a little too deep into Monty Python’s “Argument Sketch” by now?
The set { 0.9, 0.99, 0.999, … } does neither contain 0.999… nor 1 as one of its elements.
You would call me stupid, if I would claim that there are more segments in 1 than the segments [0, 0.25], [0.25, 0.5], [0.5, 0.75], [0.75, 1], wouldn’t you? And if you ask me what the additional segment is and I don’t answer - wouldn’t it be suspicious?
So, what is the segment in 0.999… which is not present in one of the numbers 0.9, 0.99, 0.999, …?
So I was right! Mr. Netz had never heard of closed sets and open sets.
What do I win?
You win? I think I pointed out that Netz didn’t know what set theory was sometime during the Bush administration. The first Bush administration.
I understand what you’re saying … my question is why you are saying this … what property of the set { 0.9, 0.99, 0.999, … } precludes 1 for being one of it’s elements …
… or are you asking me to assume first what it is you are concluding?
Oh, right, two doors down and on the left … this is the “Pointless Word Banter” sketch … so sorry about that …
That’s what we’re asking you. If .999~ <> 1.0, then you need to construct the non-zero difference. You need to develop some kind of notation to indicate what the difference consists of.
The fact is that you neither can, nor apparently will.
[0, 1] is a closed set, [0, 1) is a half-open set (point 1 excluded). So what?
Because the sequence 0.9, 0.99, 0.999, … doesn’t include its limit?
The fact is that I can’t, because 0.999… is not a defined point on the number line.