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The proof is by William A McWorter Jr.. Form the array below.
There are a+b columns and each integer appears exactly twice in the array. Now, given more than half the integers in {1,2,...,a+b}, the given integers appear in the above array, counting multiplicity, more than a+b times. Hence their must be two occurrences of the given integers in the same column. Thus those occurrences differ by a or b. The reason for the condition b<2a is that you need only a little over a third of the integers if b=2a. And the original proof still works if b>2a.
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