The variational distance between two probability distributions $X$ and $Y$ taking values on the same alphabet $\mathcal A$ is defined as \begin{equation} \delta (X,Y)=1/2\sum_{a\in A} |p_X(a)-p_Y(a)|$ \end{equation} There are two very basic claims with regard to the variational distance that I would like to formally prove.
1) It cannot increase by the application of a function:
\begin{equation} \delta (X,Y)\geq \delta (f(X),f(Y)) \end{equation}
2)
\begin{equation} \frac{1}{2}\sum_{a\in A} |p_X(a)-p_Y(a)| = 1 -\sum_{a\in A} \min (p_X(a),p_Y(a)) \end{equation}