I'm learning about cauchy sequences which is why I want to know if someone could give me a sequence $(x_n)_{n \in \mathbb{N}}$ that is not a cauchy-sequence, but $(x_{n+1}-x_n)_{n \in \mathbb{N}}$ a null sequence?
2026-03-28 08:09:37.1774685377
Sequence $(x_n)_{n \in \mathbb{N}}$ that is not a cauchy-sequence, but $(x_{n+1}-x_n)$ a null sequence
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Classic example $$ x_n = \sum_{j=1}^n \frac{1}{j}.$$ It does not converge, is thus not a Cauchy-sequence, but $x_{n+1}-x_n=\frac{1}{n+1}$ is a Cauchy sequence.