Let $x, y, z$ be three distributions on $\mathbf{N}$. Prove the properties:
$$\begin{gathered}
0 \leq d_{VT}(x, y) \leq 1 \\
d_{VT}(x, y) = 0 \Longleftrightarrow x = y \\
d_{VT}(y, x) = d_{VT}(x, y) \\
d_{VT}(x, z) \leq d_{VT}(x, y) + d_{VT}(y, z)
\end{gathered}$$