Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 224 / 1 / c

Put $p_k=P_X(k)$. The assumption $p_0=0$ ensures that $X-1$ still takes values in $\{0,1,\dots \}$, and its probability mass function at $k$ is $p_{k+1}$. Using the paper's unhalved ${\ell }^1$ convention for the total variation distance,