\begin{proof} Let $E$ be the event $\left\{ \sum_{n=1}^{\infty}Y_{n}=\infty \right\}$. For a large constant $T>0$, define the rv $$S_{N}=\min\left\{ T,\sum_{n=1}^{N} Y_{n} \right\}$$$S_{N}$ is a bounded rv, $S_{N}$ is non-decreasing, so, it has a limit and for $\omega \in E$, $$\lim_{ N \to \infty } S_{N}(\omega)=T$$By Dominated Convergence Theorem we have that $$\lim_{ N \to \infty } \mathbb{E}[S_{N}]=\mathbb{E}\left[ \lim_{ N \to \infty } S_{N} \right]=T\ge T\cdot \mathbb{P}(E)$$But, $$\begin{align*} \lim_{ N \to \infty }\mathbb{E}[S_{N}]&\le \lim_{ N \to \infty } \mathbb{E}\left[ \sum_{n=1}^{N}Y_{n} \right] \\ &= \mathbb{E}\left[ \lim_{ N \to \infty } \sum_{n=1}^{N}Y_{n} \right]\\ &= c \end{align*}$$Hence, $\mathbb{P}(E)\le \frac{c}{T}$. But $T$ is arbitrary so $T\to \infty$ yields $\mathbb{P}(E)=0$.

\end{proof}