Because the total number of handshakes is even, the sum of odd handshakes+the sum of even handshakes = even.

Even sum is even, so odd sum is even.

So * * * has an even number of odd times, which is even.

Induction:

When there is only 1 person, the number of odd-numbered handshakes is 0, which proves the proposition.

Assuming that the proposition holds when there are n people, consider the case of n+ 1

If there are an even number of handshakes in n+ 1 person, which is set to I, then I will be removed. After all the people who shake hands with I subtract 1, the remaining n people satisfy the proposition, so that the even number of handshakes still satisfy the proposition after I join;

If n+ 1 people's handshakes are all odd, then the proposition is obviously satisfied when n+ 1 is even.

When n+ 1 is odd, if one I is arbitrarily removed, the remaining n people need to meet the proposition that there are even odd handshakes, then the remaining even handshakers are the ones who shake hands with I, which is contradictory.

Then the proposition is proved, proved.