In Richard Kaye's book Models of Peano arithmetic, one can read (page 13):
We have proved that any nonstandard $M \models \mathrm{Th}(\mathbb{N})$ has a nonstandard $a \in M \models \theta(a)$ iff there are infinitely many $k \in \mathbb{N}$ satisfying $\mathbb{N} \models \theta(\underline{k})$. This observation is the basis of many elegant "non-standard" proofs of theorems about $\mathbb{N}$.
where $\theta(x)$ is a formula over the language $\mathcal{L}_A= \{0,1,+,\times ,<\}$ with only one free-variable $x$.
Do you know such elegant proofs?