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Nathan Nguyen
2024-12-06 12:01:10 -06:00
parent 74e129000b
commit 98c8b94c06
8 changed files with 29 additions and 29 deletions

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@@ -96,7 +96,7 @@ is a *subsequence* of $X$,
**Theorem**. Every sequence of real numbers $(x_n)$ contains a monotonic subsequence $(x_{n_k})$.
**Collary**. Bolzano-Weierstrass Theorem. Every bounded sequence of real numbers has a convergent subsequence.
**Corollary**. Bolzano-Weierstrass Theorem. Every bounded sequence of real numbers has a convergent subsequence.
## Section 3.5 - The Cauchy Criterion
@@ -136,9 +136,9 @@ $$
\abs{s_m - s_n} = \abs{\sum_{i = m + 1}^n x_i} < \epsilon
$$
**Collary**. $n$-th Term Test. If $\sum x_n$ converges, then $x_n \rightarrow 0$.
**Corollary**. $n$-th Term Test. If $\sum x_n$ converges, then $x_n \rightarrow 0$.
**Collary**. Absolute Convergence Test. If $\sum \abs{x_n}$ converges, then $\sum x_n$ converges.
**Corollary**. Absolute Convergence Test. If $\sum \abs{x_n}$ converges, then $\sum x_n$ converges.
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