In the textbook All of Statistics by Wasserman, there's an exercise that asks the following:
Let $X_1, X_2, \ldots$ be a sequence of random variables such that $$ \mathbb{P}\left(X_n=\frac{1}{n}\right)=1-\frac{1}{n^2} \quad \text { and } \quad \mathbb{P}\left(X_n=n\right)=\frac{1}{n^2} . $$ Does $X_n$ converge in probability? Does $X_n$ converge in quadratic mean?
This is my attempt for the first part of the question:
We denote $X_n$ as the random variable and $x_n$ as its corresponding realization. We can then write the probability of $X_n$ as $$ P(X_n = x_n) = \left(1-\frac{1}{n^2}\right) \mathbb{I}\left(x_n = \frac{1}{n}\right) + \frac{1}{n^2} \mathbb{I}(x_n = n) $$ where $\mathbb{I}$ is the indicator function. Since the probability of $X_n$ is always bounded, it follows that $X_n$ converges in probability to 0.
I'm unsure about the correctness and I'm also having trouble proceeding with the second part.