Sorry for the late reply, took me a while to (and attempt to) digest that
For your claim and proof, are you proving that
is true?
Cause I think you used that in your proof in this part:
(looks like you subbed in )
What I did is called proof by
induction. Proof by induction has three steps:
(i) Show that what you're trying to prove holds for the most basic case. This is called the base "case".
(ii) Assume that what you're trying to prove holds for all $n\le N$ where $N$ is some integer.
(iii) Using the assumption you made in (ii), show that it holds for $N+1$. If you can show that, then you have shown it holds for any value of $n$, since I can arbitrarily make $N$ as large as I want. It's a real, useful and fun proof technique. Specialist maths should include some of these proof techniques (including proof by
contrapositive).