I am trying to understand how we can show an inductive hypothesis doesn't work and must be replaced. Is there any rigorous process to this or is it just a general sense of fruitlessness with the current inductive hypothesis? I am working on a proof involving the sequence $a_1,a_2, \cdots$ defined as $a_1 = 1$ and $a_{n+1}=3a^2_n$ for $n\ge1$. The exercise I am doing is to show that if we wished to prove $a_n \le 3^{(2^n)}$, our inductive hypothesis cannot use this exact claim. I am aware of stronger claims and managed to prove this using a stronger hypothesis. What I am unsure about is how we would show a certain inductive hypothesis cannot prove something. So far, what I have in order to show the insufficiency of the inductive hypothesis using this claim is to square both sides and then multiply them by $3$, getting $3a_n^2 \le 3^{(2^{n+1}+1)}$. I have remarked that we have not gotten the claim we desired for $n+1$ here. However, I quite unsure as to whether this is a good argument, because one approach not working to show the claim for $n+1$ doesn't preclude other approaches working to show the claim using the original inductive hypothesis. So what is a good argument to show that the inductive hypothesis using $a_n \le 3^{(2^n)}$ doesn't work?
EDIT: Is it even possible for an inductive hypothesis to be impossible to prove when the claim is correct? because could we not have a proof by induction of the claim we want to prove inside of our proof by induction, this nested proof using the stronger hypothesis? Then having proved it with the stronger hypothesis inside of our proof, we could then proceed to use the original weaker claim, utilizing our nested proof with the stronger claim to argue the inductive step using the weaker claim- our stronger claim inside of the proof shows the weaker claim to be true for all $n$, so we could still assume the weaker claim to be true for some $n$ and conclude it works for $n+1$ based on our stronger claim in our nested proof?
There are two questions of concern when it comes to weaknesses in induction proofs:
The second is not the issue here, as easily seen via an example, but it can be a case where induction is not a good idea.
What about the first? We need to show that $$ a_n \le 3^{2^n} \implies a_{n+1} = 3 a_n^2 \le 3^{2^{n+1}} $$ but then, dividing by three and taking the square root (assuming the terms are all positive, which is trivially true) $$ a_n^2 \le 3^{2^{n+1} - 1} \implies a_n \le \sqrt{ 3^{2^{n+1} - 1} } = 3^{2^n - 1/2} $$ Note the issue here: we assumed that $a_n \le 3^{2^n}$, and were able to conclude that $a_n$ was even smaller. But what if $a_n$ met the first condition, but was greater than $3^{2^n - 1/2}$?
Thus, the relationship for $a_n$ does not necessarily imply a like relationship for $a_{n+1}$.