I've been asked to prove by induction that $n^2\leq 2^n$, and told it is true $ \forall n\in \mathbb{N},n>3$
I think I have found the right way to the proof, but I'm not sure since I get stuck half-way there. What I did was taking a base case of $n=4$ and tested it, and it resulted to be true. Then I assumed it would be true for some number $k$, such that $n=k$ and $k^2\leq 2^k$, and attempted to prove
$(k+1)^2 \leq 2^{k+1}$
And this is how I attempted to prove this. First of all, I started with my assumption.
$=k^2\leq 2^k$
$=2k^2\leq2^{k+1}$
Then I tried to prove that $(k+1)^2 \leq 2k^2$, for this would imply my thesis, i.e. $(k+1)^2\leq2^{k+1}$. So I went forth on my effort:
$(k+1)^2≤2k^2$
$=k^2+2k+1\leq2k^2$
$=2k+1\leq k^2$
(By assumption)
$=2k+1\leq 2^k$
Now that I simplified it, I need to prove this is true; this is, prove that $(k+1)^2 \leq 2k^2$. So I take a base case of $n=4$ and in deed it satisfies the inequality. So I assume it is true for some number $j, k=j$ and try to prove it. Nevertheless I have failed in trying to prove this, I don't really know if my steps so far are right or wrong. Is my reasoning okay? And if its, how can I prove $2k+1\leq 2^k$? Thank you in advance.