Bolzano-Weierstrass Theorem
The Bolzano-Weierstrass Theorem is a result in analysis that states that every bounded sequence of real numbers contains a convergent subsequence.
Proof: Since is assumed to be bounded we have
. Bisect the closed interval
into two intervals
and
. Let
. Take some point
. Bisect
into two new intervals, and label the rightmost interval
. Since there are infinite points in
we can pick some
, and continue this process by picking some
. We show that the sequence
is convergent. Consider the chain
![\[\ldots\subseteq I_k\subseteq I_{k-1}\subseteq\ldots\subseteq I_2\subseteq I_1.\]](http://latex.artofproblemsolving.com/e/5/3/e533ff0fc37da82371c4bde3016cd66744ca9065.png)
By the Nested Interval Property we know that there is some contained in each interval. We claim
. Let
be arbitrary. The length
for each
is, by construction,
which converges to
. Choose
such that for each
that
. Since
and
are contained in each interval, it follows that
.