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Consider the sequence $s_n=\sqrt{n}$. Although $(s_{n+1}-s_n)\rightarrow 0$ as $n\rightarrow \infty$, the sequence is not convergent (since it is unbounded).

(This is a very common example in study of Cauchy sequences where, by this example, it is remarked that making consecutive difference smaller is not sufficient for Cauchy condition.)

Suppose we put such a condition on a sequence; i.e. suppose $(x_n)$ is a sequence such that $(x_{n+1}-x_n)\rightarrow 0$ as $n\rightarrow\infty$ and $(x_n)$ is bounded. Can we say that $(x_n)$ must be convergent? If not, what condition, instead of boundedness, will guarantee the convergence of the sequence $(x_n)$?

Beginner
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    See this. – David Mitra Dec 11 '20 at 11:42
  • I added at the end, If not, what condition, instead of boundedness, will guarantee the convergence (Since I had not thought about the example in answer below, so I assed this extra point in question; I don't get why it is closing. – Beginner Dec 11 '20 at 11:53
  • In the first line of above question, I shortly mentioned why it is not Cauchy in $\mathbb{R}$ (equivalent to say, why it is not convergent). – Beginner Dec 11 '20 at 11:54
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    Also https://math.stackexchange.com/q/1494962/42969, with the example $x_n = \sin(\sqrt n)$. – Martin R Dec 11 '20 at 11:54
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    @Beginner: In a complete metric space, the convergent sequences are exactly the Cauchy sequences. Perhaps you can clarify what kind of “condition, instead of boundedness” you are thinking of. – Martin R Dec 11 '20 at 12:00

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No. Think about $$0, 1, \frac{1}{2}, 0, \frac{1}{3}, \frac{2}{3}, 1, \frac{3}{4}, \frac{1}{2}, \frac{1}{4}, 0, \frac{1}{5}, ...$$

TheSilverDoe
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  • This is certainly nice example (I would also like to see conditions under which convergence will hold). – Beginner Dec 11 '20 at 11:56
  • Nice, but not new: https://math.stackexchange.com/a/107344/42969, https://math.stackexchange.com/a/1495839/42969. – Martin R Dec 11 '20 at 11:57
  • @MartinR Yes, but it was quicker to answer the question than to search for duplicates... (I know it is bad, sorry ^^) – TheSilverDoe Dec 11 '20 at 11:58