Wait! Why do I have to use strong induction to prove the inequality holds?

@C-RAM Thanks, edited.

@Seeker Sorry about that, edited.

@cheesewiz That's all good! When asking questions such as these, it is important that you show your attempts at them. It is also discouraged to use pictures to convey text. It is completely fine to use them for visuals though. You should have a read through this post on how to use Mathjax on this website. I will edit this question for you. I will also delete my previous comment as it is not longer relevant.

@cheesewiz Before I edit, I am not sure what $I_2$ means in the question asking "use $I_2$ to prove. . ."

@Seeker I_2 is just strong induction so show it holds for n=1,2 we then assume it holds true for 1,2,.. k and then prove it holds for k+1, I edited my question again for more clarity on where I am stuck

@cheesewiz Got it!

@cheesewiz I tried to edit it as best I could. Let me know if there are any mistakes.

@Seeker could you add the value alpha which is (1+rad5)/2 it’s not clear what x is in my post

@cheesewiz Added it.

Does this answer your question? Proving that $\alpha^{n-2}\leq F_n\leq \alpha^{n-1}$ for all $n\geq 1$, where $\alpha$ is the golden ratio - found using an Approach0 search. Another duplicate is Fibonacci : demonstration by induction : $\phi^{n-2} \leq F_n \leq \phi^{n-1}$.

Did you replace $x$ by $\alpha$ just to confuse the people reading the question? Edit your question and get rid of one of them, please!

@jjagmath haha no I would never, I didn’t know how to type alpha someone else kindly edited my post for me.

Then here's an idea: use $x$

@JohnOmielan the second one helped thanks

@cheesewiz Welcome to Math SE. I'm glad that my second proposed duplicate question helped you. Also, regarding how to type "alpha", it's $\alpha$, with this showing as $\alpha$. Note the math formatting syntax used here is MathJax (quite similar to $\LaTeX$) with Short and helpful advice on using MathJax on the site ... giving a good introduction, and MathJax basic tutorial and quick reference being a reasonably comprehensive tutorial/reference.

@JohnOmielan Thank you for the reference, how would you practice it or test it out?, I can't write dummy posts on Math SE right?

@cheesewiz You're welcome. As for practicing and testing the MathJax syntax, you could write a "dummy" question, but just not press the Post button. Nonetheless, I suggest instead using one of the currently free answers in Sandbox for drafts of long, complex posts is better, as it's basically designed & set up for this sort of thing. However, although the answers there are all "community wiki", with a lower reputation threshold for editing, I don't think your reputation is high enough, so this option may not be available to you for now.