Confusion on my inductive proof of $2^n$ ≥ $n^2$ for n ≥ 4 [closed]

(The problem) Use the principle of mathematical induction to prove that $2^n$ ≥ $n^2$ for n ≥ 4

Here's my solution on paper (https://i.stack.imgur.com/iJq7M.jpg)

(1) The Basis case is true: for n = 4 we have $2^4$ = $4^2$

(2) The Induction Step: Assume $2^k$ = $k^2$ is true for any k ≥ 4

(3) Solving for the Inductive hypothesis: I deduced that $2^k$ ≥ $k^2$ ⇒ $2(2^k$ ) ≥ $2(k^2)$ and I know that $2k^2$ ≥ $(k+1)^2$. By simplifying the binomial, I got $2k^2$ ≥ $k^2+2k+1$, now, if I subtract the left side to the right side we have $k^2-2k-1$ ≥ 0 and I also get $1-2/k^2-1/k^2$ ≥ 0 by dividing $(1/k)^2$ . Which if you look at the bottom of my paper, inputting some values of k lead me to think that this claim isn't true.

Firstly, you made an error in your proof. You turned $2^{k+1}\geq(k+1)^2$ into $2k^2\geq(k+1)^2$ in the middle of the proof, which is a wrong implication. The statement you are trying to prove is true.

Secondly, even if you had made no error: failing to prove a statement is not equivalent to proving that the statement is wrong. You did not prove in any way that the statement is wrong. To do that, you would have to show that there exists a counterexample where $2^n<n^2$ for some $n\geq 4$.

Lastly, please use mathjax in the future so that your post is easily readable. Do not just photograph handwritten notes. MathJax basic tutorial and quick reference