I've been trying to find the solution for this problem for a long time... but I can't seem to do it. I'm not asking for the full solution, maybe just a hint or where to start off from. I never know where to begin with these type of problems.
Let $s \geq 1$ be a positive integer, prove that in the interval $[s,2s]$ contains a power of 2.
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Hint: Let $k \geq 0$ be an integer such that $2^k$ is the highest power of $2$ which is less than or equal to $s$. If $s=2^k$, then we are done, otherwise $2^k <s$, so what can be said about $2^{k+1}$?
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If you want greater than or less than, put %\geq% %\leq% but instead of % use $.
As to your question, I assume you mean that for all $s\in\mathbb{N}$, there exist an $n\in\mathbb{N}$ such that $s<2^n\leq2s$.
We proceed by strong induction:
Base Case: This is true for $1$ since $1<2^1\leq2$.
Strong Inductive Step: We assume that for all naturals $1,2,...s$ that the proposition is true. In order to prove this, me must show there exists $n\in\mathbb{N}$ such that $s+1<2^n\leq2(s+1)$. Now, if $s+1$ is even, then there exists $w\in\{1,2,...s\}$ and $m\in\mathbb{N}$ such that $2w=s+1$ and $$w<2^m\leq2w.$$ Multiplying be $2$, we have $$2w<2*2^m\leq2*2w$$ $$s+1<2^{m+1}\leq2(s+1)$$ and the proposition is true. Now, consider the case where $s+1$ is odd. Note that $$s<2^n\leq2s$$ is true (from our assumption). Also, since $2^n$ is even and $s$ is even, $2^n>s$ implies $2^n>s+1$. Also, using $2s+2>2s\geq2^n$, we see $$s+1<2^n<2s+2$$ $$s+1<2^n\leq2(s+1)$$ which is what we were originally trying to show.
If $s$ is a power of $2$, then there is nothing to prove. If $s$ is not a power of $2$ then it must lie between two consecutive powers of $2$, i.e., there is an integer $r$ for which $2^r < s < 2^{r+1}$. This yields $2^{r+1} < 2s$. Hence $s < 2^{r+1} < 2s$, which gives the required result.
Hope it helps.