$Y_n \overset{p}{\rightarrow} Y$ if $X_n \overset{p}{\rightarrow} X$ and $Y_n\overset{d}{=}X_n$ and $Y\overset{d}{=}X$?

Question

Suppose $X_n \overset{p}{\rightarrow} X$ and $Y_n\overset{d}{=}X_n$ and $Y\overset{d}{=}X.$ Then can we say that $Y_n \overset{p}{\rightarrow} Y$?

My thinking was to use the triangle inequality:

$P(\|Y_n-Y\|>\epsilon)\leq P(\|Y_n-X_n\|>\epsilon/3)+P(\|X_n-X\|>\epsilon/3)+P(\|X-Y\|>\epsilon/3).$

The second term goes to zero by assumption $X_n \overset{p}{\rightarrow} X$, but then I am not sure about the first and last terms since I don't assume info about joint distributions of $(X_n,Y_n)$ and $(X,Y)$.

Answer 1

Let $X\sim N(0,1)$, $X_n=X, Y_n=-X, Y=X$. This is a counter-example.