Assume, by way of contradiction, that "T"0 is not compact. Then there exists an infinite open cover "C" of "T"0 that does not admit any finite subcover. Through bisection of each of the sides of "T"0, the box "T"0 can be broken up into 2"n" sub "n"-boxes, each of which has diameter equal to half the diameter of "T"0. Then at least one of the 2"n" sections of "T"0 must require an infinite subcover of "C", otherwise "C" itself would have a finite subcover, by uniting together the finite covers of the sections. Call this section "T"1.