Cantor devised the diagonal proof of the uncountability of the real numbers:

Given a function, f, from the natural numbers to the real numbers, consider the real number r whose binary expansion is given as follows: for each natural number i, r's i-th digit is the complement of the i-th digit of f(i).

Thus, since r and f(i) differ in their i-th digits, r differs from any value taken by f.

Therefore, f is not surjective (there are values of its result type which it cannot return).

Consequently, no function from the natural numbers to the reals is surjective.

A further theorem dependent on the axiom of choice turns this result into the statement that the reals are uncountable.

This is just a special case of a diagonal proof that a function from a set to its power set cannot be surjective:

Let f be a function from a set S to its power set, P(S) and let U = x in S: x not in f(x).

Now, observe that any x in U is not in f(x), so U != f(x); and any x not in U is in f(x), so U != f(x): whence U is not in f(x) : x in S.

But U is in P(S).

Therefore, no function from a set to its power-set can be surjective.