I'm new to induction and trying to prove $2^n > n$ for all natural numbers.

I've seen a couple of examples but am confused about the the case going from $k = 1$ to $k =2$.

So I show $2^1 > 1$ as the base case.

Then I assume $2^k > k$

Meaning that

$2.2^k > 2k$

i.e.

$2^{k+1} > 2k$

Or

$2^{k+1} > k + k > k + 1$

So it is considered proven.

But when $k = 1$, $k + k \not> k + 1$

What am I missing please?

Do I need a special case for going from $k=1$ to $k=2$?