A doubt about greatest integer function

Question

This is a question with reference to a previous post.
Inequalities on greatest integer function

In the question posted there, I understand why the options A, B, D are true or false. But there is no strong proof that option C, $[2^x] \leq 2^{[x]}$ is true or false. So can you please explain about that option C.

In that post only a counterexample is given. I want solid algebraic proof.

Answer 1

If $[x]=k$ then $k \leq x <k+1$ which implies $2^{k } \leq 2^{x} <2^{k+1}$. This implies $l\leq 2^{x} <l+1$ for some $l \geq 2^{k}$. Hence $[2^{x}]=l \geq 2^{k}=2^{[x]}$.