Answer
By definition, a perfect square is an integer that equals the square of some integer. Therefore, we wish to prove that there are integers $k$, $m$, and $n$ such that $k^{2} = m^{2} + m^{2}$. Let $k=5$, $m=4$, and $n=3$. Then $k^{2}=5^{2}=25=16+9=4^{2}+3^{2}=m^{2}+n^{2}$. Since our choice of $k$, $m$, and $n$ satisfies the hypothesis and conclusion, we conclude that there is a perfect square ($25$) that can be written as the sum of two other perfect squares ($16$ and $9$).
Work Step by Step
This proof is a constructive proof of existence, whereby we show that something exists by finding a specific example. For more on this method of proof, see the section entitled "Proving Existential Statements" beginning on page 148, especially example 4.1.3.
As a side note, the statement this problem asks us to prove is equivalent to the statement that the Pythagorean Theorem has an integer solution. Thus, any of the Pythagorean triples from elementary geometry (3-4-5, 5-12-13, and so on) will work for this proof.
