Example of a series, which its square sum converges, but it is not a Cauchy sequence.

Question

Give an example of a sequence $\{x_{n}\}$, in any normed vectorial space, such that $$\sum_{n\ge1}\|x_{n}\|^{2}\le\infty $$ but when you put $$S_{n}=\sum_{i=1}^{n}x_{i}$$ the sequence $\{S_{n}\}$ is not a Cauchy sequence. I would appreciate if someone could help me with this.

Answer 1

Assuming $\leq$ means $<$, then $x_n=1/n$ is the standard example.

Answer 2

Assuming $\|x_n\|$ just means absolute-value and $x_n$ are real numbers:

If the series converges absolutely, then for any $\varepsilon > 0$ there is an associated $N$ for which $$\sum_{n=N}^{\infty} |x_n| < \varepsilon$$

Then for any $m \geq n \geq N$ we have $$|S_n - S_m| = \left| \sum_{p=n}^m x_n \right| \leq \sum_{p=n}^m |x_n| \leq \sum_{n=N}^{\infty} |x_n| < \varepsilon.$$