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, given a bounded sequence 
, 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 
.
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.
