 triangle inequality and the easily proved fact that, for any state distribution $\mu \in \cP(\sX)$ and any two Markov kernels $P,P' \in \cM(\sU|\sX)$, $$\begin{aligned}     \left\| \mu K(\cdot|P) - \mu K(\cdot|P') \right\|_0 \le \max_{x \in \sX} \left\| P(\cdot|x) - P'(\cdot|x) \right\|_0.\end{aligned}$$ Letting now the initial state distribution be $\mu_0$, we can apply the bound recursively to obtain $$\begin{aligned} \left\| \mu