A sequence $(a_n)$ where $\exists M>0$ such that $\forall n\in\mathbb{N}$, $\sum\limits_{k=1} ^n |a_{k+1}-a_k|\leq M$. Show $(a_n)$ is Cauchy sequence

Question

Question:

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence where there exists $M>0$ such that for all $n\in\mathbb{N}$, $\sum\limits_{k=1} ^n |a_{k+1}-a_k|\leq M$. Show that $(a_n)_{n\in\mathbb{N}}$ is a Cauchy sequence and thus convergent.

Hint:

First show that the sequence $(b_n)_{n\in\mathbb{N}}$ defined by $b_n=\sum\limits_{k=1} ^n |a_{k+1}-a_k|$ for all $n\in\mathbb{N}$ is convergent and thus Cauchy.

So far I have followed the hint through:

Consider the sequence $(b_n)_{n\in\mathbb{N}}$ where $b_n=\sum\limits_{k=1} ^n |a_{k+1}-a_k|$ for all $n\in\mathbb{N}$. We know there exists $M>0$ such that for all $n\in\mathbb{N}$, $\sum\limits_{k=1} ^n |a_{k+1}-a_k|\leq M$, and thus $(b_n)_{n\in\mathbb{N}}\leq M$ for all $n\in\mathbb{N}$. We also have that: $$ b_{n+1}=\sum\limits_{k=1} ^{n+1} |a_{k+1}-a_k|=\left(\sum\limits_{k=1} ^n |a_{k+1}-a_k|\right)+|a_{n+2}-a_{n+1}|=b_n+|a_{n+2}-a_{n+1}| $$ and thus $b_{n+1}\geq b_n$ for all $n\in\mathbb{N}$. Hence $(b_n)_{n\in\mathbb{N}}$ is both bounded above and monotone increasing and thus by the Monotone Convergence Theorem, $(b_n)_{n\in\mathbb{N}}$ is convergent, and so $(b_n)_{n\in\mathbb{N}}$ is a Cauchy sequence.

However, I don't know how I'm supposed to apply this to the main part of the question. Is $(b_n)_{n\in\mathbb{N}}$ a subsequence of $(a_n)_{n\in\mathbb{N}}$?

Answer 1

The sequence $c_n=\sum_{k=1}^n(a_{k+1}-a_k)$ is also Cauchy. $$ c_n=(a_2-a_1)+(a_3-a_2)+\dots+(a_{n}-a_{n-1})+(a_{n+1}-a_{n})=a_{n+1}-a_1. $$

Answer 2

Consider a sequence $\{ b_n\}$ where $ b_n:=\sum_{k=1}^n |a_{k+1}-a_k| $. Then $$ b_{n+1} - b_n =|a_{n+2}-a_{n+1} | \geq 0 $$ So it is increasing. And since $b_n \leq M$ for all $n$, so it is bounded. Hence $\{b_n\}$ converges. Clearly $b_n$ is Cauchy. That is, for $\varepsilon >0$, there exists $n_0$ s.t. $$ n > m>n_0 \Rightarrow |b_n - b_m |< \varepsilon $$ So $$ \varepsilon > b_n-b_m = \sum_{k=m+1}^n |a_{k+1} -a_k| \geq | \sum_{k=m+1}^n (a_{k+1} -a_k) | = |a_{n+1}-a_{m+1} | $$ which implies that $\{a_n\}$ is Cauchy.