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Let $(a_n)$ be a sequence in $\mathbb{R}$ and let $b_n = a_{n+1} - a_n $ be a zero sequence. According to my intuition, I would say that $a_n$ converges. But my solution says otherwise. How could that be? I just can't find a counterexample.

Shaun
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Anna Saabel
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    A zero sequence? Is that one that converges to zero? Like say $\sqrt{n+1}-\sqrt n$? – Angina Seng Jul 25 '18 at 10:51
  • Another one: https://math.stackexchange.com/q/1019832/42969. – Martin R Jul 25 '18 at 11:00
  • You have that a sequence converges if it's cauchy, since this one is a special case of cauchy $m=n+1$, one would expect that there exists such sequence so that there exists a $m$ such that $a_n-a_m$ doesn't go to $0$. Intuitively, a sequence which diverges slowly will satisfy $a_{n+1}-a_n\to 0$. – kingW3 Jul 25 '18 at 11:09
  • Counterexample $a_{n}=\frac{1}{n}$ – Dr. Wolfgang Hintze Jul 25 '18 at 13:10

3 Answers3

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$$a_n = \sum_{k=1}^n \frac 1 k$$

Kenny Lau
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Counterexample: $a_n=1+\frac{1}{2}+...+\frac{1}{n}$.

Fred
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Another nice example. $a_n = \sqrt{n}$. Show that $$ \sqrt{n+1} - \sqrt{n} \to 0\qquad\text{but}\qquad \sqrt{n} \to +\infty $$

GEdgar
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