Let $\mathcal{C} = \{x_1, x_2, \cdots\}$ be a countable set of $\mathbb{R}$. Let $\{\mathbb{P}_n\}$ and $\mathbb{P}$ be probability measures on $\mathcal{C}$. Prove that $\mathbb{P}_n \stackrel{w}{\longrightarrow} \mathbb{P}$ if and only if $\mathbb{P}_n(x_k) \longrightarrow \mathbb{P}(x_k)$ for every $k \in \mathbb{N}$, and $\sum_{k=1}^\infty \mathbb{P}(x_k) = 1$.
The if part is easy to prove (In fact, available on MSE, e.g., here). However, I'm stuck while proving the only if part which demands that the weak convergence of $\{\mathbb{P}_n\}$ to $\mathbb{P}$ to imply pointwise convergence of each of them at each $x_k$. Essentially, we are required to show that for any continuous and bounded function $f$, $$\sum_{k = 1}^\infty f(x_k) g_n(x_k) \longrightarrow 0\quad\implies\quad g_n(x_k) \longrightarrow 0\quad \forall k \in \mathbb{N}$$ where $\{g_n\}$ is a sequence of functions bounded between $-1$ and $1$, defined as $g_n(x) = \mathbb{P}_n(x) - \mathbb{P}(x)$. I couldn't show this.
Please correct me if I'm wrong. Any help will be appreciated.