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Bolzano-Weierstrass Theorem
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.