The sequence $(y_n)$ has to be unbounded, not bounded, otherwise there are obvious counterexamples like $a_k = 0$ for all $k$.

I'll suppose that $y_n$ should be unbounded. Assume wlog that $\limsup y_n = \infty $, then there exist infinitely many $m \in \mathbb{N}$ such that for some $n \in \mathbb{N}$, $y_n < m \le y_{n+1}$ (indeed, take $n+1 = \min \{ k \colon y_k \ge m \}$). Furthermore, we have $a_k \to 0$ and thus $y_{n+1} - y_n = a_{n+1} < \frac{1}{3}$ for $n$ sufficiently large. Therefore $\{ y_{n+1} \} < \frac{1}{3}$ and $\{ y_n \} > \frac{2}{3}$ for sufficiently large $n$ as above, which means that $\{ y_n\}$ cannot converge.