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What is a sequence $\{a_n\}$ that diverges and $\displaystyle\lim_{n\to 0} |a_n-a_{n+1}| =0$

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Take $$ a_n=\sqrt{n} $$ then note that, for $n\ge1$, $$ a_{n+1}-a_n=\sqrt{n+1}-\sqrt{n}=\frac1{\sqrt{n+1}+\sqrt{n}}. $$

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Hint: consider harmonic series.

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  • $\begingroup$ This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review $\endgroup$ – Björn Friedrich Dec 11 '16 at 9:13
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    $\begingroup$ @BjörnFriedrich "This does not provide an answer to the question." Actually it does, very much so. $\endgroup$ – Did Dec 11 '16 at 9:26
  • $\begingroup$ @BjörnFriedrich It very much hints to an answer of the question. Perhaps adding the words 'the partial sums sequence of" between "consider" and harmonic" makes this more direct, but hints are very welcome. $\endgroup$ – DonAntonio Dec 11 '16 at 9:57
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Besides the two good answers you already got, you could also take

$$a_n:=\log n\implies a_{n+1}-a_n=\log\left(1+\frac1n\right)\xrightarrow[n\to\infty]{}\;\ldots$$

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