Average of a periodic voltage
While the other answers are correct, personally I have difficulty 'seeing' a proof's correctness from formulae. So let's get a little more hand-wavy.

Let's assume the average of a single period of the waveform is known. The average of the next period will be the same, this is the definition of period after all, each period is the same. 

Given this, if we average over exactly one period, or two periods, or any integer number of periods, then we will get the same average.

However, what happens if we average over half a period? We'll get a different answer. So averaging over 1.5, or 2.5, or 10.5 periods will all give different answers, as the contribution from the half period is different to the contribution from the full period.

But, there is a trend. If we average over n.5 periods, we will have n contributions from the consistent average, and only one contribution from the different one. As n becomes larger, the contribution from the half period has a smaller effect, by the factor of n. As we allow n to grow without bound, the half period contribution shrinks to become a negligible part of the average. In the limit, as n tends to infinity, we drop all this 'tending to' language, and just say that they are equal, that the average is undisturbed by any non-period contributions.