The Sliding Boxes Proof


Theorem: A monotone function on [a,b] is integrable.

Suppose f is increasing on [a,b]. Then for each positive integer n, the left Riemann sum LEFT(n) is an underestimate for the lower integral of f over [a,b], while RIGHT(n) is an overestimate for the upper integral of f over [a,b]. The difference between of the upper integral minus the lower integral of f over [a,b] is then bounded below by 0 and above by RIGHT(n)-LEFT(n). By sliding boxes, one can see that RIGHT(n)-LEFT(n) is a constant times 1/n. Hence, the upper and lower integrals of f over [a,b] must agree.