Let’s say what you have is a *sequence*, and it is defined *recursively*. We can apply *Banach*’s theorem about fixed points.

starting point: a(1) = root(2)

recursion: a(i) = root(2) ^ a(i-1)

In order to apply the theorem, we will show the following:

a = 2 is a fixed point (obviously: root(2) ^ 2 = 2)

a(i) will grow towards 2: a(i) > a(i-1) (true, because when we raise x to the power of y, and x>1, y>1 then x^y>x)

a(i) will not grow beyond 2: a(i) < 2 (true, because the power needed to raise root(2) beyond 2 is greater than 2, so once smaller than 2, forever smaller than 2)

That’s it! We proved that the limit of your sequence is 2.

Oh yes: The value 4 is also a fixed point for this recursive definition: root(2) ^ 4 = 4, but you will need a different starting point. With the information given, you may be able to consider different ranges of starting points and their limits.