A sequence that satisfies the condition: $\forall p \in \mathbb{N}: \lim_\limits{n \to \infty } | a_p - a_{p+1}| = 0$ but is not a cauchy sequence.

Question

I need to find a sequence that satisfies the condition: $\forall p \in \mathbb{N}: \lim_\limits{n \to \infty } | a_n - a_{n+p}| = 0$ but is not a cauchy sequence.

This somehow implies for me that I need a convergent sequence to satisfy the condition, however this contradicts the second condition of the task, namely that an is not a Cauchy sequence.

So there clearly is something else I'm not taking into account but I can't figure out what. Please help me.

Your condition is very confusing; you take a limit over $n$ but $n$ doesn't appear anywhere in that expression. — Steven Stadnicki, May 07 '20 at 18:52
It's still a bit confusing; $\lim_p$ 'binds' the variable $p$, so the condition ;'for all $p$' doesn't really make sense. Presumably you just mean the same condition but without the $\forall$ clause before it, that is to say that $\lim_p |a_p-a_{p+1}|=0$? — Steven Stadnicki, May 07 '20 at 18:54
Also: https://math.stackexchange.com/q/2374955/42969, https://math.stackexchange.com/q/3052948/42969, https://math.stackexchange.com/q/2741224/42969, https://math.stackexchange.com/q/3346676/42969 and some more – all found with Approach0 — Martin R, May 07 '20 at 18:58

score 2 · Accepted Answer · answered May 07 '20 at 18:53

2

Define $$a_n =\sum_{k=1}^n \frac{1}{k}$$ Clearly, $\{a_n\}$ isn't convergent but it satisfies the condition you want.

answered May 07 '20 at 18:53

user6

4,092

A sequence that satisfies the condition: $\forall p \in \mathbb{N}: \lim_\limits{n \to \infty } | a_p - a_{p+1}| = 0$ but is not a cauchy sequence.

1 Answers1