Can someone explain how the induction step is carried out in this proof?

Question

I do not understand the part where it says that as $k \geq 3$ we can go from $2^k + 2$ to $2^k + 2^k$.

Answer 1

He said this reason for the step before this one, i.e. he meant that as $k\ge 3$ then from our assumption in the induction we have $2k+1 \le 2^k$.

But the step after it is always true, I mean $2 \le 2^k$ for any $k\ge 1$ (which is the same as saying $2^k+2\le 2^k+2^k$)

Answer 2

$2\leq 2^k$ for all $k\geq 3$.

In fact, this is true for all $k\geq1$, but we have specifically chosen $k$ to be 3 or more in this proof.