The Bolzano-Weierstrass theorem states that every bounded infinite set in 
has an accumulation point.
For 
,
an infinite subset of a closed bounded set 
has an accumulation point in 
. For instance, choose a bounded
sequence 
of distinct points in that subset, with 
for all 
. It must have a monotonic subsequence 
. The subsequence 
must converge because it is monotonic
and bounded. Because 
is closed, it contains the limit of 
, which is an accumulation
point of that subset.
The Bolzano-Weierstrass theorem is closely related to the Heine-Borel theorem and Cantor's intersection theorem, each of which can be easily derived from either of the other two.
