How do we go about proving the following part of Slutsky's theorem?
If $X_n \xrightarrow{d} X,\quad Y_n \xrightarrow{P} c$, then $X_nY_n \xrightarrow{d} Xc$ where $c$ is a degenerate random variable.
I tried using the following fact:
If $|X_n-Y_n| \xrightarrow{P} 0, \quad Y_n \xrightarrow{d} Y$, then $X_n \xrightarrow{d} Y$.
However, I could not arrive at a continuous transformation to use this fact.
I tried by lim sup and lim inf approach, directly from the definition of convergence in distribution:
$X_n \xrightarrow{d} X$, if $F_n(x) \rightarrow F(x)$ at all points of continuity of $F$, where $F_n$ and $F$ are the distribution functions of $X_n$ and $X$ respectively.
Is there any other equivalent characterisation that can help me proving Slutsky's theorem?