We've already explained above why this should be true: as the box shrinks, the function f becomes constant on it -- the box only contains points at which f takes on pretty much just one value, say the value in the given corner.

So pull this constant value outside the integral. Doing the integral gives the volume. But then we divide this out, and are just left with the value of f at the corner of the box.

If you want to see a proof, one is provided, but follow the train of thought through to the end before checking it out -- if you do happen to feel like seeing how a formal proof looks.

Note that this is exactly the Divergecne Theorem for the special case of a cube. That is, the speacial case of a cube is the key to the genreal case.