Any n-cube is compact.

Definition:

A closed n-cube in can be described by a pair where is the "-th" corner and is the length of a side. For example if then the corners of the closed square are

Using this notation, one can define a subdivision process that corresponds to bisecting and interval. Below, we only give the details for the square.

Definitions:

1. Given a closed square in we can subdivide it into 4 closed squares of area . In our notation the squares are:

2. In the obvious way, one can also define a notion of a closed square nested in a closed square . Note that each of the subsquares in 1. of are nested in .

Theorem:

Let be an countable sequence of nested closed squares, then

Proof(Sketch)

If one argues that we have sequences

...bounded above by

and

... ...bounded above by

Hence each sequence has a least upper bound.

Letting be the pair. One argues as before that since all the squares are closed sets.