Inductive Proof, is this way valid?

Have to prove 2^n >= n^2 for n>=4

I.H.: Works when I plug in four. Show that 2^(n+1) >= (n+1)^2 or show that 2^n * 2 >= (n+1)^2

Since we know 2^n >= n^2 then 2^n * 2 >= 2n^2

So I then go about showing through another induction proof that 2n^2 >= (n+1)^2

...2(n+1)^2 >= (n+2)^2

2n^2 + 4n + 2 >= n^2 + 2n + 4

n^2 + 4n + 2 >= 2n + 4

n^2 + 2n + 2 >= 4

Since we know n must be >= 4 ...16 + 8 + 2 >= 4

So we add to what we know... 2^n >= 2n^2 >= (n+1)^2 therefore 2^n >= (n+1)^2.

Is this valid? Thanks.