If $X_n\rightarrow^{p}\mu_{n}$ and $\mu_{n}\rightarrow\mu$, does $X_n\rightarrow^{p}\mu$ as well? [closed]

Solution 1:

There is a problem with writing $$\lim_{n\to\infty}P(|X_n-\lim_{n\to\infty}\mu_n|>\epsilon)=0$$ which is that the $n$ in the outer limit and the $n$ in the inner limit are two different dummy variables. Each is only defined with the confines of its own limit. But because one limit is inside the other the $n$ in $\mu_n$ is ambiguous. Is it the $n$ of the inner limit, or of the outer limit? In this case, it isn't hard to figure out, but you shouldn't have to figure it out.

The correct usage is to have different variable names: $$\lim_{n\to\infty}P(|X_n-\lim_{m\to\infty}\mu_m|>\epsilon)=0$$

I'm sure that spoils your next step, which I guess would have been to say $$\lim_{n\to\infty}P(|X_n-\mu_n|>\epsilon)=0$$

But this is something you need to justify in a more rigorous fashion.