Earlier this morning, I had posted another answer (now deleted). But on my way to the chocolate shop I've realized it answered a different (and much easier) question I must have dreamed of... 

In addition, and what is perhaps more interesting, I remembered that Georges Grekos told me of work by T. Šalát on the very question in the OP. Here is a reference:

> T. Šalát, *On subseries*, Math. Zeitschr. **85** (1964), 209-225.

There, Šalát proves, among other things, the following:

> **Theorem.** Let $(a_n)_{n \ge 1}$ be a non-increasing sequence of non-negative real numbers such that $a_n \to 0$ as $n \to \infty$ and $\liminf_n n a_n > 0$. If $(\varepsilon_n)_{n \ge 1}$ is a $\{0,1\}$-valued sequence for which $\sum_{n \ge 1} \varepsilon_n a_n < \infty$, then $\lim_n \frac{1}{n} \sum_{k=1}^n \varepsilon_n = 0$.

This is Theorem 1 in Šalát's paper, which also investigates the logical strength of the assumptions of the previous result. In particular, Note 2 on p. 211 shows that, at least in general, you can't replace the hypothesis that $\liminf_n n a_n > 0$ in the above thereom with the weaker condition that $\sum_{n \ge 1} a_n = \infty$. For this, Šalát considers the sequence $(a_n)_{n \ge 1}$ defined by letting $a_{n^n + k} := n^{-(n+2)}$ for all $n, k \in \bf N$ such that $0 \le k < (n+1)^n - n^n$.