If $\{a_n\}$ is a convergent sequence of real numbers, then $ \lim_{n \to \infty} n(a_{n+1} - a_n) = 0$

Question

The question is as follows:

If $\{a_n\}$ is a convergent sequence of real numbers, then $\lim_{n \to \infty} n(a_{n+1} - a_n) =0 $.

I have been thinking about this question since yesterday but I am unable to prove it. $a_{n+1} - a_n \rightarrow 0$ if $\{a_n\}$ is convergent, but there is also a term of $n$ which is creating trouble for me. I verified this with examples like $a_n = 1/n$ and $a_n =1$ but can't get any idea on how to prove it. Please help me, any idea will be appreciated.

@Gary Can you repost the link to the question you suggested to look at? I would like to have a second look at it and I am unable to search for it. Thanks — P-addict, Jun 22 '21 at 08:13
See https://math.stackexchange.com/questions/2509422/convergent-sequence-and-lim-limitsn-to-inftyna-n1-a-n-0?rq=1 — Gary, Jun 24 '21 at 06:50

score 3 · Accepted Answer · edited Jun 24 '21 at 05:57

3

This is false. Take $a_n= \sum\limits_{k=1}^{n}\frac {(-1)^{k}} {k}$

edited Jun 24 '21 at 05:57

Joe

19,636

answered Jun 22 '21 at 08:00

Kavi Rama Murthy

311,013

Thanks a lot, Sir. – P-addict Jun 22 '21 at 08:01

If $\{a_n\}$ is a convergent sequence of real numbers, then $ \lim_{n \to \infty} n(a_{n+1} - a_n) = 0$

1 Answers1