Prove that given a nonnegative integer $n$, there is a unique nonnegative integer $m$ such that $(m-1)^2 ≤ n < m^2$ 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?

Its too late to answer the question but if it helps you can prove it by contradiction also.
Assume that there exists a k such that k is less than m. 
so 
(k−1)2≤n< k2
The smallest k which is possible is k = m-1. Then we have 
(m-2)2 ≤ n< (m-1) 2 
which is contradicting the assumed statement. so the solution has a unique value.
