Bounded Monotone Convergence Theorem

Theorem

1. If $\{a_n\}$ is Bounded
2. If $\{a_n\}$ is Monotone
3. In particular, it is either:
   1. Strictly Increasing and Bounded Above
   2. Strictly Decreasing and Bounded Below Then, $\{a_n\}$ Converges

Proof

Proving Strictly Increasing and Bounded Above Case

1. Suppose $\{a_n\}$ is Bounded Above
2. Suppose $\{a_n\}$ is Strictly Increasing
3. Then, $\exists c\in \mathbb{R}$ s.t $\forall a\in \{a_n\},a\le c$
4. Then, $\forall x_1,x_2\in \mathbb{N},x_1<x_2⟹a_{x_1}<a_{x_2}$
5. We want to show $\exists l\in R,\forall ϵ>0,\exists N>0$ s.t $n>N⟹|a_n-l|<ϵ$
6. Consider $A=\{a_n|n\in \mathbb{N}\}\subset \mathbb{R}$
7. Note that $a_1\in A$ so $A$ is non-empty
8. Note $A$ is bounded above, as we know $\{a_n\}$ is bounded above
9. By the Completeness Axiom, the Supremum of the set $A$ exists
10. Choose $l=sup(A)$
11. Then, choose arbitrary $ϵ>0$
12. Choose $N\in \mathbb{N}$ s.t $l-ϵ<a_N$ by definition of supremum
13. Suppose $n>N$
14. Note that $l-ϵ<a_N<a_n$ as $\{a_n\}$ is strictly increasing
15. Note by definition that since $l=sup(A),\forall a\in A,a<l$
16. Note $a_n\in A$ by definition
17. Then, $a_n\le l<l+ϵ$
18. Then, by transitivity of $<$, it follows that $l-ϵ<a_n<l+ϵ$
19. $⟹-ϵ<a_n-l<ϵ$
20. $|a_n-l|<ϵ$ by $|\cdot |$ defn
21. Thus, we have shown convergence
22. $\square$

Examples

• Proving Convergence with BMCT