I have a question on irreducible Markov Chains that has been bugging me for a few hours now I have the markov chain defined by:
$P(i, i-1) = 1 - P(i,i+1) = \frac{1}{2(i+1)}$ for $i>=1$, and $p(0,1) = 1$.
Now this chain is irreducible and I'm asked to prove that a.s. starting from state $i$ we hit the state $a$ when $a > i$ in a finite amount of time. I think it can be proven by saying that for all states between $0$ and $a$, we have a probability $ p > (a+2) /2*(a+1) > 0.5$ to do +1, so I think I can compare the markov chain to a random walk of uniform probability $\frac{a+2}{2(a+1)}$ which tends towards $+\infty$.
Is my reasoning correct? I feel that either there is a much more direct method or that what I'm trying to prove is true in general as long as the chain is irreductible (regardless of the transition probabilities)
Thanks!