Answer
Let $m=1$ and $n=-1$. Then $\frac{1}{m}+\frac{1}{n}=\frac{1}{1}+\frac{1}{-1}=1-1=0$. Since $0$ is an integer, we conclude that there are distinct integers $m$ and $n$ such that $\frac{1}{m}+\frac{1}{n}$ is an integer.
Work Step by Step
This proof is called a constructive proof of existence, whereby we show that something exists by finding a specific example. For more on this, see the section "Proving Existential Statements" beginning on page 148, especially example 4.1.3.
Be careful not to confuse this problem with problem 4. This problem is about distinct integers (integers that do not equal one another), whereas problem 4 is about integers greater than $1$ that are not necessarily distinct.
