_Posted on May 14, 2024_ > [!theorem] **Theorem:** _Bounded monotone sequences converge._ Proof: Suppose we have $(a_n)$, a bounded increasing sequence. Let $S = \{a_n\mid n \in \mathbb{N}\}$, and let $u = \sup S$. We want to show that $\lim_{n\rightarrow \infty} = u$. Let $\epsilon > 0$. By the definition of $\sup$, $u - \epsilon$ is not an upper bound for $S$. Hence, there exists an $N$ such that $a_N > u - \epsilon$. Since $(a_n)$ is increasing, $n > N$ implies that $u - \epsilon < a_n < u$ which implies that $|a_n - u| < \epsilon$ and so $\lim_{n \rightarrow \infty}(a_n) = u$ The proof for decreasing sequences is similar $\blacksquare$ > [!theorem] **Theorem:** _Absolutely convergent series converge._ Proof: Suppose that we have a series $\sum|b_n|$ which converges. We can partition the sequence $(b_n)$ into two subsequences $(b_n^+)$ and $(b_n^-)$ where $b_n^+ = \max(b_n, 0)$ and $b_n^-= \min(b_n, 0)$. Note that $\sum b_n = \sum b_n^+ + \sum b_n^-$. Thus we want to show that both $\sum b_n^+$ and $\sum b_n^-$ converge so that by additivity of convergent series we obtain our result. Notice that both of the corresponding sequences of partial sums are monotone. Now, we must show that they are bounded. Since the sequence $(\sum_{k=m}^n |b_k|)$ converges, it is bounded, so there exists and $M$ such that $0 \leq \sum_{k = m}^n b_k^+ \leq \sum_{k=m}^n |b_k| \leq M$ and by linearity of convergent sequences, $-M \leq -\sum_{k=m}^n|b_k| \leq \sum_{k=m}^n b_k^- \leq 0.$ Thus, both $\sum b_n^+$ and $\sum b_n^-$ converge. By additivity of convergent series, $\sum b_n = \sum b_n^+ + \sum b_n^-$ converges $\blacksquare$ There is an alternate proof of this result using the [[Comparison Test]]. However, we use the fact that absolutely convergent series converge in our primary proof of the comparison test. Thus I will eventually ( #TODO ) provide another proof of the comparison test in that post, as well as given the alternate proof of this results here. Proof (using comparison test): --- The proof that bounded monotone sequences converge was adapted from _Elementary Analysis_ by Kenneth A. Ross