# Prove that $\ \lim_{m \to \infty} \mathbb{P}(T_0 < \infty|X_0 = m) = 0 \$ where $T_0$ is the waiting time to reach state $0$.

Let $$(X_n)_{n \geq0}$$ be a Markov Chain on {$$0,1,...$$} with transition probabilities given by: $$P_{0,0}=1, \ \ P_{i,i+1}= p_i, \ \ P_{i,i-1}= q_i, \$$ for all $$i \geq 1$$, and $$P_{i,j}=0 \$$ otherwise.

Let $$T_i = \text{min}$${$$n \ge 0 \ : \ X_n=i$$}.

I was reading through a problem in my text with the above information given and found that the fact$$\ \lim_{m \to \infty} \mathbb{P}(T_0 < \infty|X_0 = m) = 0 \$$, was used in a proof but there was no justification as to why it was true. Intuitively this makes sense because as your initial state becomes larger and larger, each 'path' to state $$0$$ has a lower and lower probability of happening. But I can't seem to prove this rigorously.