Prove using induction:
For any natural number $n$ there is a natural number $m$ such that $n\le m^2\le 2n$.
Obviously letting $n$ and $m$ equal $1$ satisfies the first part of mathematical induction. I'm stuck at the second part. I believe we assume the inequality holds for $n=k$ but I am stuck on where to go next. I know we have to prove the inequality holds for $k+1$ but am not sure how to go about that.