Two Positive Real Numbers $m$ and $m$ such that $m<n$ and $\sqrt{m}<\sqrt{n}$

Question

This is a homework problem I came across for my discrete math class and I can't manage to figure out the answer. I know it is asking for an example where $m$ and $n$ are positive real numbers, $m$ is less than $n$, and $\sqrt{n}$ is less than or equal to $\sqrt{m}$.

Find a counterexample to the following statement: “If $m$ and $n$ are positive real numbers and $m < n$, then $\sqrt{m} < \sqrt{n}$."

Answer 1

Since $$\sqrt{m}-\sqrt{n}=\frac{(\sqrt{m}-\sqrt{n})(\sqrt{m}+\sqrt{n})}{\sqrt{m}+\sqrt{n}}=\frac{m-n}{\sqrt{m}+\sqrt{n}}<0,$$ we have no the needed counterexample.

Answer 2

By squaring the second inequality (between positive numbers),

$$m<n\land\sqrt m\ge \sqrt n\implies m<n\land m\ge n\ !$$