Let $\Delta(X,Y)$ denote the (variational) distance between two discrete random variables $X$ and $Y$. $\Delta(X,Y)$ is given by: $\Delta(X,Y)=\frac{1}{2}\sum_{v \in V} |\Pr[X=v] - \Pr[Y=v]|$. $V$ is the set of possible values for $X$ and $Y$. We can also see $\Delta(X,Y)$ as the distance between the distributions of $X$ and $Y$. Thus $\Delta(X,Y)=0$ means that the distributions of $X$ and $Y$ are the same.
Suppose that two positive integers $M$ and $K$ are given, with $M\leq K$. And let $X$ be a random variable that takes on values in $\{0,\ldots,M-1\}$, and $Y$ a random variable uniform on $\{0,\ldots,K-1\}$. Then the following holds: $\Delta(Y,X+Y)\leq \frac{M-1}{K}$.
I wrote it out several times but I don't get how they came up with that bound. Can anyone help me with this?