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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.

  1. $\{X_n-Z\}$

  2. $\{X_n-W\}$ 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.

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Hint for (2): Write out $X_n - W$ in terms of $Z$, $Y_n$ and $W$. What random variable should it be very close to when $n$ is large?

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