The number of odd-handshakers is always even

At a party, people shake hands with each other in some pattern (some pairs shake, some don't; nobody shakes their own hand). Count each person's total handshakes. Some people will have shaken hands an even number of times, others an odd number of times.

Prove that the number of people who shook hands an odd number of times is always even, no matter how the handshaking happened.