# If $Z_n\overset{d}{\rightarrow}Z,\ Y_n\overset{d}{\rightarrow}a$ then $Y_nZ_n\overset{d}{\rightarrow}aZ$ [duplicate]

If $$Z_n\overset{d}{\rightarrow}Z,\ Y_n\overset{d}{\rightarrow}a$$ where $$a\in\mathbb{R}$$ where the CDF of Z is continuous then $$Y_nZ_n\overset{d}{\rightarrow}aZ$$

I tried proving using a similliar proof to Slutsky's theorem with sum and got that if I prove $$\mathbb{P}(|Z_n(Y_n-a)|>\delta)\rightarrow 0$$
I finish.

I know that $$Y_n \overset{\mathbb{P}}{\rightarrow}a$$ So I feel like I am close to the answer, I havn't used the fact that the CDF of Z is continuous yet, so I feel like I'm supposed to use it here.

Is there a way to complete the proof?
If not, can I get a hint how to solve the problem and how to use the fact that the CDF of Z is continuous.

Thank you,

• The hypothesis that $Z$ have no point masses is not needed. Jun 27 at 15:35
• Yes, many thanks Jun 27 at 16:06
• Now that I tried to prove it, I got stuck at showing that $|๐๐(๐๐โ๐)|\rightarrow 0$ weak convergence. The problem presented in the link needs probability convergence, but I wonder if its possible to prove weak convergence. Jun 27 at 16:52