----- > [!proposition] Proposition. ([[monotone convergence theorem for sequences]]) > Let $(a_{n})_{n \in \mathbb{N}}$ be a [[monotonic map|monotone]] [[sequence]] of real numbers. Then $(a_{n})$ [[sequence|converges]] if and only if it is bounded, and in this case: $\lim_{n \to \infty}a_{n}=\begin{cases} \sup_{n} a_{n } \text{ if } a_{n} \leq a_{n+1} \text{ for all } n \\ \inf_{n} a_{n} \text{ if } a_{n} \geq a_{n+1} \text{ for all } n. \end{cases}$ ^proposition > [!proof]- Proof. ([[monotone convergence theorem for sequences]]) > > First note that any convergent monotonic sequence is bounded by its limit. Indeed, suppose $(a_{n}) \to L$ is increasing but has $a_m > L$ for some $m$. Because increasing, $a_{n}\geq a_{m}>L$ for all $n > m$. Choose $N>m$ such that $\underbrace{ |a_{n}-L| }_{ =a_{n}-L }<a_{m}-L$ for all . Then $a_{n} -L< a_{m}-L \text{ for all }n \geq N, $ > implying $a_{n}<a_{m}$ for all $n \geq N$. This is a contradiction. The argument for the decreasing case is analogous. Thus, monotonic + converges implies bounded. > > For the converse: suppose $(a_{n})$ is a monotonic sequence (say, increasing) that is bounded above. We claim that $(a_{n})$ [[sequence|converges]] to its [[supremum]] $c:=\sup_{n}a_{n}$. Fix $\varepsilon>0$. Then there exists $N$ with $c-\varepsilon < a_{N} < c$; otherwise $c-\varepsilon$ would be a smaller upper bound than $c$. Thus $\varepsilon>c-a_{N}$. For $n \geq N$, then, we have $a_{N} \leq a_{n} < c$ and $|c-a_{n}|=c-a_{n} \leq c-a_{N}<\varepsilon.$ > Analogous argument for the decreasing case. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```