Must there be a sequence $(\epsilon_n)$ of signs such that $\sum\epsilon_nx_n$ and $\sum\epsilon_ny_n$ are both convergent?

Let $(x_n)$ and $(y_n)$ be real sequences.

(i) Suppose $x_n \rightarrow 0$ as $n \rightarrow \infty.$ Show that there is a sequence $(\epsilon_n)$ of signs (i.e., $\epsilon_n \in \{−1, +1\}$ for all $n$) such that $\sum \epsilon_nx_n$ is convergent.

(ii) Suppose $x_n \rightarrow 0$ and $y_n \rightarrow 0.$ Must there be a sequence $(\epsilon_n)$ of signs such that $\sum\epsilon_nx_n$ and $\sum\epsilon_ny_n$ are both convergent?

I'm struggling to come up with formal proofs, for (i) I've seen that we simply pick a limit and and then as soon as our sum passes the limit we set $\epsilon_n=-1$ until we pass it again and so on, oscillating about the limit but as $x_n \rightarrow 0$ we converge to it. for (ii) I don't think there must be such a sequence of $\epsilon_n$ but I can't construct a proof or counter example. So I would ask for a solution to (ii) and possibly a better way of constructing answers/tackling these problems in general.

Thank you

The answer to (ii) is yes, there is such a sequence of signs $\epsilon_n$. See Theorem 2.2.1 here, where this result (for any number of series, formulated for vector valued series) is referred to as the Dvoretzky-Hanani Theorem.