Let a_{1},...,a_{k} and b be *positive* real numbers. Then the equation
a_{1}/`x`_{1} + ... + a_{k}/`x`_{k} = b

has only a *finite* number of solutions in natural numbers `x`_{1},...,`x`_{k}.

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.