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**Theorem**. Every sequence of real numbers $(x_n)$ contains a monotonic subsequence $(x_{n_k})$.
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**Collary**. Bolzano-Weierstrass Theorem. Every bounded sequence of real numbers has a convergent subsequence.
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**Corollary**. Bolzano-Weierstrass Theorem. Every bounded sequence of real numbers has a convergent subsequence.
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## Section 3.5 - The Cauchy Criterion
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@@ -136,9 +136,9 @@ $$
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\abs{s_m - s_n} = \abs{\sum_{i = m + 1}^n x_i} < \epsilon
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$$
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**Collary**. $n$-th Term Test. If $\sum x_n$ converges, then $x_n \rightarrow 0$.
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**Corollary**. $n$-th Term Test. If $\sum x_n$ converges, then $x_n \rightarrow 0$.
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**Collary**. Absolute Convergence Test. If $\sum \abs{x_n}$ converges, then $\sum x_n$ converges.
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**Corollary**. Absolute Convergence Test. If $\sum \abs{x_n}$ converges, then $\sum x_n$ converges.
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---
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