Let $(X_t)_{t \ge 0}$ be a finite irreducible and aperiodic Markov chain with state space $\Omega$, transition matrix $P$ and stationary distribution $\pi$. Let $\| \cdot \|$ denote the total variation distance and define
$c(t)=\max_{x \in \Omega}\|{1 \over t}\sum_{s=0}^{t-1}P_x^s - \pi\|$
$d(t)=\max_{x \in \Omega}\|P_x^t-\pi\|$
Suppose $t_m$ is the mixing time according to the Cesaro metric $c(t)$, and $s_m$ the mixing time according to the total variation distance $d(t)$. Show $t_m({1 \over 4}) \le 6s_m({1 \over 8})$, where $t_m(\epsilon)$ implies $c(t_m(\epsilon)) \le \epsilon$.
I've learn many results that touch this problem, but I can't link them together to do the proof. I know we can define $p(t)=\max_{x,y \in \Omega}\|P_x^t-P_y^t\|$ and that $d(t) \le p(t)$. Also, $p(kt) \le p(t)^k$. Coupling can be used to bound total variation distance. But I don't see a coupling that will be usefull here.
This is exercise 10.12 from Markov Chains and Mixing Times, 2009, Levin, Perez, Wilmer.