Consider exercise 1.3.4 of Norris' Markov Chains. The question is as follows:
Let $\{X_n\}_{n\geq 0}$ be a Markov Chain with state space $S=\{0,1,2,\dots\}$. Suppose the transition probabilities are given by $$p_{01}=1,\,\,\,p_{i,i+1}+p_{i,i-1}=1, \,\,\,p_{i,i+1}=\dfrac{(i+1)^2}{i^2}p_{i,i-1}$$ for $i\geq1$. Show that $P[X_n\geq1\space\forall n\geq1\mid X_0=0]=\dfrac{6}{\pi^2}$
I defined $h_k=P[X_n\neq 0\space\forall n\geq1\mid X_0=k]$ for all $k\geq0$. Note first that $h_0=h_1$ so it is enough to compute $h_1$. I have deduced that for any $k\geq1$, $$h_k=h_1S_k \quad \text{ where }\quad S_k=\sum_{i=1}^k\dfrac{1}{i^2}$$
I will be done if I can show that $\lim_{k\to\infty}h_k=1$. (And in fact, this will be the case). However, I cannot show it. Intuitively it is clear, but I cannot really show it rigorously.
I noted that $h_k$ are increasing in $k$, so the limit $\lim_{k\to\infty}h_k$ exists. But why should this limit be $1$?