I'm struggling to understand the proof of the following theorem:
Theorem: Let $\big\{\mathsf X\big\}_{n\in\mathbb N}$ be an homogeneous DTMC, P its transition matrix and $\mathsf S$ its states space; let $i,j\in\mathcal S\mid i \leftrightarrow j$, then $f_{ij} = 1$
Proof: Let $m\in\mathbb N\;\lvert\; P^{(m)}_{ji} > 0$. Then
$$1 = f_{jj} = \mathbb P\big( \mathsf X_n = j \text{ for infinitely many } n\mid \mathsf X_0 = j \big) \\= \mathbb P\big( \mathsf X_n = j \text{ for some } n > m+1 \mid \mathsf X_0 = j \big) = \cdots$$
And so on, I won't report the remaining part cause this one right here is the only one giving me a bad headache.
Now, why is the third equality true? I mean, to me the events in the probability argument sound totally different. I saw alternative proves involving renewal processes but we didn't study them in our class and aren't part of our program, I need to understand this one right here.
EDIT: Supposedly the two arguments should be equivalent, i.e.
$$\mathsf X_n = j \text{ for infinitely many } n \iff \mathsf X_n = j \text{ for some } n > m+1$$
The "$\implies$" part is obvious, but I can't see why the "$\impliedby$" holds.
We saw and proved a theorem asserting that if a state $k\in\mathsf S$ is recurrent then paths starting in $k$ are a.c visiting $k$ at least once ($f_{kk} = 1$) and then if $k$ is recurrent the paths starting in $k$ return a.c. infinitely many times in $k$, may this have something to deal with it?
$\def\F{\mathscr{F}}\def\peq{\mathrel{\phantom{=}}{}}$The property used in the equality is actually:\begin{gather*} P(X_n = j \text{ for infinitely many } n \mid X_0 = j) = 1\\ \Longleftrightarrow P(X_n = j \text{ for some } n > m + 1 \mid X_0 = j) = 1. \end{gather*} For $\Longrightarrow$, it is obvious by$$ \{X_n = j \text{ for infinitely many } n\} \subseteq \{X_n = j \text{ for some } n > m + 1\}. $$ For $\Longleftarrow$, define$$ τ_0 = 0, \quad τ_{n + 1} = \inf\{k > τ_n + m + 1 \mid X_k = j\}, \quad \forall n \geqslant 0 $$ then$$ \{X_n = j \text{ for infinitely many } n\} = \bigcap_{n = 1}^∞ \{τ_n < ∞\} = \lim_{n → ∞} \bigcap_{k = 1}^n \{τ_k < ∞\}. $$ So it suffices to prove by induction on $n$ that$$ P(τ_1 < ∞, \cdots, τ_n < ∞ \mid X_0 = j) = 1. \quad \forall n \geqslant 1 $$ For $n = 1$,$$ P(τ_1 < ∞ \mid X_0 = j) = P(X_n = j \text{ for some } n > m + 1 \mid X_0 = j) = 1. $$ Now assume that it holds for $n$. Denote $A_k = \{τ_1 < k - m - 1, \cdots, τ_{n - 1} < k - m - 1\}$ for $k \geqslant m + 2$, then$$ \{τ_1 < ∞, \cdots, τ_n < ∞\} = \bigcup_{k = m + 2}^∞ (A_k \cap \{τ_n = k\}) $$ and\begin{align*} &\peq P(τ_1 < ∞, \cdots, τ_n < ∞, τ_{n + 1} < ∞ \mid X_0 = j)\\ &= \sum_{k = m + 2}^∞ P(A_k \cap \{τ_n = k\} \cap \{τ_{n + 1} < ∞\} \mid X_0 = j)\\ &= \sum_{k = m + 2}^∞ \sum_{l = k + m + 2}^∞ P(A_k \cap \{τ_n = k\} \cap \{τ_{n + 1} = l\} \mid X_0 = j). \tag{1} \end{align*} Note that $\{τ_n = k\} \subseteq \{X_k = j\}$, so\begin{align*} &\peq P(A_k \cap \{τ_n = k\} \cap \{τ_{n + 1} = l\} \mid X_0 = j)\\ &= P(A_k \cap \{τ_n = k\} \cap \{X_k = j\} \cap \{τ_{n + 1} = l\} \mid X_0 = j)\\ &= P(A_k \cap \{τ_n = k\} \cap \{X_k = j\} \mid X_0 = j)\\ &\peq · P(τ_{n + 1} = l \mid A_k \cap \{τ_n = k\} \cap \{X_k = j\}). \end{align*} By the Markov property,\begin{align*} &\peq P(τ_{n + 1} = l \mid A_k \cap \{τ_n = k\} \cap \{X_k = j\}) = P(τ_{n + 1} = l \mid X_k = j)\\ &= P(X_{k + 1} ≠ j, \cdots, X_{l - 1} ≠ j, X_l = j \mid X_k = j)\\ &= P(X_1 ≠ j, \cdots, X_{l - k - 1} ≠ j, X_{l - k} = j \mid X_0 = j)\\ &= P(τ_1 = l - k \mid X_0 = j), \end{align*} combining with$$ P(A_k \cap \{τ_n = k\} \cap \{X_k = j\} \mid X_0 = j) = P(A_k \cap \{τ_n = k\} \mid X_0 = j) $$ and the induction hypothesis yields\begin{align*} (1) &= \sum_{k = m + 2}^∞ \sum_{l = k + m + 2}^∞ P(A_k \cap \{τ_n = k\} \mid X_0 = j) P(τ_1 = l - k \mid X_0 = j)\\ &= \sum_{k = m + 2}^∞ \sum_{l = m + 2}^∞ P(A_k \cap \{τ_n = k\} \mid X_0 = j) P(τ_1 = l \mid X_0 = j)\\ &= \left( \sum_{k = m + 2}^∞ P(A_k \cap \{τ_n = k\} \mid X_0 = j) \right) \left( \sum_{l = m + 2}^∞ P(τ_1 = l \mid X_0 = j) \right)\\ &= P(τ_1 < ∞, \cdots, τ_n < ∞) P(τ_1 < ∞ \mid X_0 = j)\\ &= P(τ_1 < ∞, \cdots, τ_n < ∞) = 1. \end{align*} End of induction.