Understanding induction proof with inequalities I'm having a hard time proving inequalities with induction proofs. Is there a pattern involved in proving inequalities when it comes to induction? For example:
Prove ( for any integer $n>4$ ): 
$$2^n > n^2 \\ $$
Well, the skipping ahead to the iduction portion, here's what I've got so far:
let $n=k$, then $2^{k+1} > (k+1)^2 $. 
Starting with the LHS 
$$\begin{align}     \\  2^{k+1}=2\cdot2^k  \\  > 2\cdot k^2  \end{align}$$
And that 's where it ends for me. What would be the next step, and what suggestions might you give for future proofs like this? Do we always need to prove that the left hand side is equal to the other?

Suppose $2^n > n^2$. Then $2^{n+1}=2\cdot 2^n > 2n^2$. We are done if we can show that $2n^2>(n+1)^2$ for $n>4$. This is true iff $n^2 > 2n + 1$, which it is for $n>4$. (One way to prove this might be to note that the derivative of $x^2-2x-1$ is positive for $x\geq 4$, therefore it is increasing, and it takes a positive value at 4.)
