Prove that given a nonnegative integer $n$, there is a unique nonnegative integer $m$ such that $(m-1)^2 ≤ n < m^2$
My first guess is to use an induction proof, so I started with n = k = 0:
$(m-1)^2 ≤ 0 < m^2 $
So clearly, there is a unique $m$ satisfying this proposition, namely $m=1$.
Now I try to extend it to the inductive step and say that if the proposition is true for any $k$, it must also be true for $k+1$.
$(m-1)^2 + 1 ≤ k + 1 < m^2 + 1$
But now I'm not sure how to proof that. Any ideas?