Could anyone help with a question I got from my homework?
The question is as follow: "let $Y_n$ be a sequence of random variables with mean $0$ and variances $1/n$, and let $Z\in\mathcal{N}(0,1)$. Define $X_n=Z+Y_n$. Show whether the following converges in probability and identify the distribution of the limit.
$\{X_n-Z\}$
$\{X_n-W\}$ where $W$ has the same distribution as $Z$ but is independent of $X_n$."
I tried the first one with Chebyshev's inequality and got it converges to $0$ (not sure if it's right). But I have no idea with the second. Thanks in advance.