The handshake lemma

Let $d_i$ be the number of hands shaken by person $i$. Each handshake involves two people, so
$$\sum_id_i=2h,$$
where $h$ is the total number of handshakes. This sum is even.

Modulo $2$, each $d_i$ is $0$ or $1$, so the sum is congruent to the number of people with $d_i$ odd. Since the sum is even, that number is even.