It is a question in Durrett's book probability: theory and examples, Chapter 6, Exercise 6.6.3.
Assume the state space is finite. For any transition matrix $p$, define $\alpha_n = \sup_{i,j} \frac{1}{2}\sum_{k}|p^n(i,k)-p^n(j,k)|.$ The $1/2$ is there because for any $i$ and $j$ we can defince random variables $X$ and $Y$ so that $P(X=k)=p^n(i,k), P(Y=k)=p^n(j,k)$, and $P(X \neq Y)=\frac{1}{2}\sum_{k}|p^n(i,k)-p^n(j,k)|. $
Show that $\alpha_{m+n}\leq\alpha_{n}\alpha_{m}$.
Anyone has any idea? It seems that I can easily prove that $\alpha_{m+n}\leq 2\alpha_{n}\alpha_{m}$, but I just cannot improve more.