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Every man who has lived on earth has made a certain number of handshakes. Prove that the number of men who have made an odd number of handshakes is even.

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the total number of handshake participations by all men what so ever is an even number, which is twice the number of handshakes.
the sum of all participations by men having an even number of handshakes is an evenn number, which is the sum of several even numbers. The sum of all participations by men having an odd number of handshakes is an even number, which is an evenn number minus an even number. the number of men having an odd number of handshakes must be even for the sum of the odd numbers of their participiations be even.
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