Strangely enough, this isn't actually true! Ok, so it is true that if you were to go through one by one and label some real numbers with natural number, you would never run out. But the interesting thing comes when you start with the assumption that you have labelled every real number with a natural number. Turns out there's always some left over...
It's only a proof if you take a leap of faith and accept certain things. You could argue that that's how math works, but to me the less axioms we need, the better.
I really love this quote from Wittgenstein:
"Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one".
There is a famous proof about this by Georg Cantor: http://en.wikipedia.org/wiki/Cantors_diagonal_argument