In this question, we fix a natural number $k$. Determine
$$y_k = \lim_{n \rightarrow +\infty} P_n\left(X_n = k\right).$$
Let $Y$ be a random variable on a probability space $(\Omega, \mathcal{A}, P)$, taking values in $\mathbf{N}$, and satisfying
$$\forall k \in \mathbf{N} \quad P(Y = k) = y_k.$$
Identify the distribution of $Y$.