Answer
By definition, an integer $k$ is prime if and only if $k=rs$, where $r$ and $s$ are integers, implies that either $r=1$ and $s=k$ or $r=k$ and $s=1$. Let $n=0$. Then $n^{2}-5n+2$$=0^{2}-5(0)+2=2$. Since the only positive integers $r$ ans $s$ satisfying $2=rs$ are $1$ and $2$, it must be that $2$ is prime. Since $0$ is an integer, $n=0$ satisfies both the hypothesis and the conclusion, so we conclude that there is an integer $n$ such that $n^{2}-5n+2$ is prime.
Work Step by Step
This 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.
Note that, although the choice $n=0$ might seem trivial, there is nothing wrong with it. Although there other values that would also work for this proof (for example, $n=5$), there is no need to find a more complicated value when a simpler one works just as well.
