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 there 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.