The Pigeonhole Principle (or Dirichlet's box principle) is a method introduced usually quite early in the mathematical curriculum. The examples where it is usually introduced are (in my humble experience) usually rather boring and not too deep.
It is well-known, however, that there are great and deep applications of it in research mathematics.
What applications of the pigeonhole principle would you consider in an "Introduction to proofs" course for university students? They should be non-trivial but accessible for undergraduate students, and an interchange between different mathematical fields is always welcome.