If a sequence $\{X_n\}_{n\in\Bbb N}$ satisfies $X_{n+1} - X_n \to 0$, must $\{X_n\}_{n\in\Bbb N}$ be a Cauchy sequence, thus convergent?

Asked Jan 30 '16 at 21:00

Active Jan 30 '16 at 21:06

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If a sequence $\{X_n\}_{n\in\Bbb N}$ satisfies $X_{n+1} - X_n \to 0$, must $\{X_n\}_{n\in\Bbb N}$ be a Cauchy sequence, thus convergent?

Intuitively, if a sequence is convergent (Cauchy), then as $n$ gets larger, the difference between two terms will shrink to $0$. But I feel that $\{X_n\}_{n\in\Bbb N}$ needn't to be Cauchy if $X_{n+1} - X_n \to 0$. I have no idea how to approach it.

edited Jan 30 '16 at 21:06

asked Jan 30 '16 at 21:00

Tanny Liu

2

Of course, no. Let $x_n=\sum\limits_{k=1}^n \frac1k$ – Jan 30 '16 at 21:03

If a sequence $\{X_n\}_{n\in\Bbb N}$ satisfies $X_{n+1} - X_n \to 0$, must $\{X_n\}_{n\in\Bbb N}$ be a Cauchy sequence, thus convergent?

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