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.
Contact the MathWorld Team
Bolzano's theorem
