Slutsky's theorem for central limit theorem

Question

Assume $\sqrt{n}(X_n-X)\overset{d}{\to} N(0,\sigma^2)$ and $Y_n\overset{p}{\to} c$, where $X$ is a random variable and $c$ is a positive constant. Does it hold that $\sqrt{n}(Y_nX_n-cX)\overset{d}{\to} N(0,c^2\sigma^2)$?