Let $(X_n)_n$ be a sequence of random variables (not necessary defined on the same probability space) converging in distribution to $X$, and let $(u_n)_n$ and $(y_n)_n$ be 2 sequences of real numbers converging respectively to $u_0$ and $y_0.$
Prove, using the characteristic functions, that $(X_nu_n+v_n)_n$ converges in distribution to $u_0X+y_0$.
In this problem, Slutsky's lemma is useless, because $(X_n)_n$ aren't defined on the same probability space.
The main problem is how to prove that $\forall x \in \mathbb{R},\lim_n\varphi_{X_n}(u_nx)=\varphi_X(u_0x).$
We have $|\varphi_{X_n}(u_nx)-\varphi_X(u_0x)| \leq |\varphi_{X_n}(u_nx)-\varphi_{X_n}(u_0x)|+|\varphi_{X_n}(u_0x)-\varphi_X(u_0x)|$
How can we prove that $|\varphi_{X_n}(u_nx)-\varphi_{X_n}(u_0x)|$ converges to 0.
Let $(x,y,w) \in \mathbb{R}^3,$
$$|\varphi_{(X_n,u_n,y_n)}(x,y,w)-\varphi_{(X,u_0,y_0)}(x,y,w)|=|e^{i(u_ny+wy_n)}\varphi_{X_n}(x)-e^{i(u_0y+wy_0)}\varphi_X(x)|\leq|e^{i(u_ny+y_nw)}(\varphi_{X_n}(x)-\varphi_{X}(x))|+|\varphi_{X}(x)(e^{i(u_ny+y_nw)}-e^{i(u_0y+wy_0)})|$$
and the limit is $0,$ we conclude with the continuous mapping theorem