Theorem
Every bounded closed interval
is countably compact.
Proof
A subset
of a topological space
is countably compact if every finite subset
has an accumulation point in
The Bolzano - Weierstrass Theorem states that every bounded infinite set of real numbers contains an accumulation point.
Thus
has an accumulation point
Sinceis cl;osed and
the accumulation point
ofbelongs to
Henceis countably compact.