If positive real numbers,then
Further if then
Every bounded sequence of reals contains a convergent subsequence.
Lemma 1: A bounded increasing sequence of reals converges to its least upper bound.
Proof: Suppose is a bounded increasing sequence of reals, so . Let . We want to show that . Assume . By the definition of convergence, we need to produce some such that for all , . Since is the least upper bound of the sequence, there exists an such that , and since is increasing, we know that for all , . Therefore for all , . This shows that converges to . This completes the proof of Lemma 1.
We can prove analogously that a bounded decreasing sequence of reals converges to its greatest lower bound.
Lemma 2: Every sequence of reals has a monotone subsequence.
Proof: Given a sequence of reals, we consider two cases:
- Case 1: Every infinite subsequence of contains a term that is strictly smaller than infinitely many later terms in the subsequence.
In this case, we build a strictly increasing subsequence .
Note that we can apply the assumption of Case 1 to itself. Thus there exists some such that there exist infinitely many such that . Let , and consider the subsequence where , and is the th term in after that is greater than . This exists by the assumption.
Now we can apply the assumption of Case 1 to . Thus there exists some such that there exist infinitely many such that . Since is a subsequence of , there exists an such that . We then let , and let be a subsequence of such that , and is the th term in after that is gerater tha . Note that is a subsequence of .
We can continue in this fashion to construct a strictly increasing subsequence of .
- Case 2: Case 1 fails.
In other words, there exists a subsequence of such that every term in is at least as large as some later term in . In this case, we can build a decreasing subsequence .
Consider such a sequence . Let . By the assumption of Case 2, there exists some . Let . By the assumption again, there exists some . Let . We can continue in this fashion to build a decreasing subsequence of . Since is a subsequence of , we have that is a decreasing subsequence of . This completes the proof of Lemma 2.
The Bolzano-Weierstrass Theorem follows immediately: every bounded sequence of reals contains some monotone subsequence by Lemma 2, which is in turn bounded. This subsequence is convergent by Lemma 1, which completes the proof.
This article is a stub. Help us out by.