Show that if $X_n$ and $Y_n$ are independent random variables for $1 \le n \le \infty$, $X_n \Rightarrow X_{\infty}$, and $Y_n \Rightarrow Y_{\infty}$, then $X_n + Y_n \Rightarrow X_{\infty} + Y_{\infty}$. Where $\Rightarrow$ means converge weakly or converge in distribution.
The solution as following
Since $X_n \Rightarrow X_{\infty}$ and $Y_n \Rightarrow Y_{\infty}$, we have $\varphi_{X_n}(t) \to \varphi_{X_\infty}(t)$ for all t, and $\varphi_{Y_n}(t) \to \varphi_{Y_\infty}(t)$ for all $t$ by the continuity theorem. And thus $$ \varphi_{X_n + Y_n} (t) = \varphi_{X_n} (t) \varphi_{Y_n}(t) \to \varphi_{X_\infty}(t)\varphi_{Y_\infty}(t){\color{Red} =} \varphi_{X_\infty + Y_\infty}(t)$$ for all $t$. Since both $\varphi_{X_\infty} (t)$ and $\varphi_{Y_\infty}(t)$ are continuous at $0$, $\varphi_{X_\infty + Y_\infty} (t)$ is continuous at $0$. Therefore by continuity theorem we have $X_n + Y_n \Rightarrow X_\infty + Y_\infty$.
I don't understand why $\varphi_{X_\infty}(t)\varphi_{Y_\infty}(t){\color{Red} =} \varphi_{X_\infty + Y_\infty}(t)$ for all $t$.If we can illustrate $X_\infty$ and $Y_\infty$ are independent,then the equality holds clearly.But how to prove it from $X_n$ and $Y_n$ are independent.
The claim is false. Let $X_n\sim \mathcal{N}(0,1),Y_n\sim \mathcal{N}(0,1)$, independent $\forall n$ and $X\sim \mathcal{N}(0,1),Y=-X$. Then $X_n \stackrel{\textrm{w}}\to X$, $Y_n \stackrel{\textrm{w}}\to Y$ but $X_n+Y_n\stackrel{\textrm{w}}{\to}Z$ where $Z\sim \mathcal{N}(0,2)$ and $X+Y=0$