In fact I think it can be generalised to any number k of variables:
FACT: Let a_1 .. a_k be distinct integers. Then \sum [a_i - a_(i+1)]^2 >= 4k - 6 .
You can prove it by induction on k now. The case k=2 is clear. Then if there is a counter-example to the case k, you can easily make it into a counter-example to the case k-1 . (By obliterating the largest integer.)
So this is really induction in the style of Fermat's method of infinite descent.