Help with a pigeonhole principle?

Question

Let $n \geq 1$ be an integer. Use the Pigeonhole Principle to prove that in any set of $n + 1$ integers from $\{1, 2, . . . , 2n\}$, there are two integers that are consecutive (i.e., differ by one).

Answer 1

Hint: Each set $\{1, 2\}, \{3, 4\}, ..., \{2n - 1, 2n\}$ contains zero, one, or two elements of the set. If one of the sets contains two integers in our given set, we're done; if not, bad things happen.