Let $(X_n)_{n\in\mathbb{N}_0}$ be an irreducible and recurrent Markov chain with state space $E$ and transition matrix $P$. For an $i\in E$ let $t(i)$ denote the random variable $t(i)\colon\Omega\to\mathbb{N}\cup\left\{+\infty\right\}, \omega\mapsto\inf\left\{n\geq 1: X_n(\omega)=i\right\}$.
Now it is to show that for any $i,j\in E$ it is $$ \mathbb{P}_j(t(i)<\infty):=\mathbb{P}(t(i)<\infty|X_0=j)=1. $$
In our reading, we had a rather long (and complicated) way to show that which I will add if you want to see it.
But I wonder why the following which I have on my mind is not working:
Because of the irreducibility there is a $m\in\mathbb{N}$ such that $$ p_{ji}^{(m)}:=\mathbb{P}_j(X_m=i)>0. $$ Isn't $\mathbb{P}_j(t(i)<\infty)=\mathbb{P}_j(\exists n\in\mathbb{N}: X_n=i)$?
And so because the above $m$ exists the probability is 1?
In particular, I do not see where recurrence is needed.