If $x$ and $y$ are two probability distributions on $\mathbf{N}$, we define the application $x * y : \mathbf{N} \rightarrow \mathbf{R}_+$ by
$$\forall k \in \mathbf{N} \quad (x * y)(k) = \sum_{i=0}^{k} x(i) y(k-i) = \sum_{i+j=k} x(i) y(j)$$
Show that $x * y$ is a distribution on $\mathbf{N}$.