Does Cauchy Convergence Criterion for a series imply absolute value of sequence be monotonically decreasing?

Question

Does the Cauchy Convergence Criterion for a series imply the absolute value of the sequence be monotonically decreasing? If \begin{align} S = \lim_{n \rightarrow \infty} \sum_{k=1}^{n} x_k \end{align} then $|x_{k+1}| < |x_k|$? I think the answer is yes, since $\lim_{n\rightarrow \infty} x_n = 0$, but I have not seen this written in my simple analysis textbook and I am having a little bit of a difficult time proving it (or coming up with a contradiction).

Answer 1

Look for example at the series $$\frac{1}{2^2}+\frac{1}{1^2}+\frac{1}{4^2}+\frac{1}{3^2}+\frac{1}{6^2}+\frac{1}{5^2}+\cdots.$$ This converges to the same thing, namely $\frac{\pi^2}{6}$, as its more famous cousin. But the sequence $\frac{1}{2^2},\frac{1}{1^2}, \frac{1}{4^2}, \frac{1}{3^2}, \dots$ is not monotone. The sequence, is, however, Cauchy, and the sequence of partial sums is also Cauchy.