If $\sum x_n$ converges, is the sequence $a_k =\sum_{n=1}^k x_{2n}^2$ Cauchy?
I suspect that it isn't but a bit stuck on finding a counter example. For a valid counterexample, I believe I need a series that converges but the squared even partial sum does not.
Writing the definition of Cauchy, we know a sequence $(x_n)$ is Cauchy if for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that $m, n \geq N \implies |x_m -x_n| < \epsilon$. But it seems easier to just use that convergence implies Cauchy rather than work directly from the definition. Any ideas/tips on what exactly I'm not seeing?
Hint: Let $\{y_n\}$ be a sequence of positive reals that converges to $0$, and take
$$x_n=\begin{cases} -y_{n/2} & \mathrm{if\ }n\equiv 0\bmod 2 \\ y_{(n+1)/2} & \mathrm{if\ }n\equiv 1\bmod 2,\end{cases}$$
or in other words $x_1,x_2,\cdots,$ is $y_1,-y_1,y_2,-y_2,y_3,-y_3\cdots$.
Then, your second series is
$$\sum_{n=1}^k y_n^2.$$
Can you find a sequence $\{y_n\}$ that converges to $0$ for which the above does not converge?