Consider a Markov Chain with finitely many transition states. Show there is $M>0$ and $\alpha<1$ such that $p_{ij}^{(n)}\leq M\alpha^n$ for all $n$, whenever $i,j$ are transient states.
I was thinking of considering $\displaystyle \sup_{i,j \text{ transient}}p_{ij}=\alpha$. Now I tried to do induction but that didn't seem to work out. $$p_{ij}^{(n+1)}=\sum_{k\in S}p_{ik}^{(n)}p_{kj}$$ I tried to utilise this, but to no avail. Can someone help me out? Thanks a lot.