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I recently decided to learn real analysis, and started using Krantz's analysis text. I was working through the second chapter on sequences and series and came across this question:

Let $\{a_j\}$ be a sequence. Suppose for all $N>M>0$, $|a_M-a_{M+1}|+|a_{M+1}-a_{M+2}|+\ldots+|a_{N-1}-a_N|\leq 1$. Prove that $\{a_j\}$ converges.

First of all, I know that this question is a duplicate of Proof that if the sum of the differences of consecutive terms of a sequence between any two terms is bounded above by one, then the sequence converges. However, I don't quite understand the accepted answer in that question. I want to prove this using only the definition of convergence and Cauchy sequence, either by directly choosing an $N>0$ for an arbitrary $\epsilon>0$, or through a contradiction, as these are the only thing I've learned so far. When I try to prove by contradiction it just feels like I'm running around in circles, and when I try to prove it directly I haven't the faintest idea how to choose $N$. Any help would be greatly appreciated.

Also, I know I'm not suppose to open a new question when there's already a duplicate. If there's something else I could have done, please do inform me in the comment section. Thank you.

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Since you are a new learner in this topic, I avoid using the term of "series". Hopefully this answer is clear to you.

Let $S_n=\sum_{k=1}^n|a_k-a_{k+1}|$ for $n\in\mathbb N$. By hypothesis, $S_n\leq 1$ and $S_n$ is increasing, so it is convergent.

Given any $\epsilon>0$, there exists $N>0$ such that for any $n>m>N$, we have that $S_n-S_m<\epsilon$, which means $|a_{m+1}-a_{m+2}|+\cdots+|a_{n}-a_{n+1}|<\epsilon.$ By triangle inequality, we have that $|a_{n+1}-a_{m+1}|<\epsilon$, so $\{a_n\}$ is a Cauchy sequence, as desired.

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  • $\begingroup$ Thank you! That was very well explained. $\endgroup$ – Aden Dong Jul 22 at 23:55
  • $\begingroup$ @AdenDong You are welcome. $\endgroup$ – Feng Shao Jul 22 at 23:56

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