If $\sum\limits^{\infty}_{n=1}|x_n|^2<\infty $ then $\{x_n\}$ is a Cauchy sequence

Question

Let $\{x_n\}$ be a sequence of real numbers. How do I prove that if $\sum_{n=1}^\infty|x_n|^2<\infty$ then $\{x_n\}$ is a Cauchy sequence ?

Is the converse true?

How do I prove this?

For the converse, is $\{x_n\}=1+\frac{1}{x_n}$ working?

Answer 1

If $\displaystyle\sum_{n=1}^{+\infty}\vert x_n\vert^2<+\infty$, then $\lim \ \lvert x_n\rvert^2=0$, hence $\lim x_n=0$, so $\{x_n\}$ is convergent, hence it's a Cauchy sequence.

If $x_n=\dfrac 1{\sqrt{n}}$ for all $n>0$ then $\{x_n\}$ is convergent to $0$, hence is a Cauchy sequence, but $\displaystyle\sum_{n=1}^{+\infty}\lvert x_n\rvert^2=+\infty$ (it's the harmonic series). So the converse is false.