In the analysis book I am studying, in the chapter about the real number field, there is the standard question about the intersection of nested intervals (link1), and it is (unlike in link1) presented as the intersection of infinitely many intervals: $\cap_{i=1}^{\infty} A_i $ with Ai being the closed intervals.
I first wrote a proof by induction, which is very simple and possible, but it turns out that a proof by induction will prove something else, namely that this holds true up to 'any' natural $\cap_{i=1}^{n}A_i$, but it does not prove what is requested.
Now I definitely agree that what I proved is something else, and I can intuitively see why, so the question is not about that.
The question is: What is the formal/rigorous explanation of the exact difference between what an induction proof proves, and what is being requested, which can only be proved using the 'completeness axiom' (or equivalent).
(I am aware of the seemingly similar question, but it was quite vague and was therefore closed)