Let a1,...,ak and b be positive real numbers. Then the equation
a1/x1 + ... + ak/xk = b
has only a finite number of solutions in natural numbers x1,...,xk.
This theorem is intuitively obvious, yet surprisingly difficult to prove! However, it has a beautiful concise proof using non-standard analysis.
Since writing the above, I've worked out a "standard" proof of Sierpinski's theorem. It is interesting to compare and see how more complicated a standard life is.