Space $=S=\{0,1,2,\cdots\}$
Transition probabilities $P(n,n+1)=p, P(n,0)=1-p$
$T_n$ is the first time the Random Variable returns to $n$
I want to show that $P_n(T_n<\infty)=P(T_n<\infty|X_0=n)=1\,\,\forall n\in\mathbb N$
I was able to show $P_0(T_n<\infty)=P_n(T_0<\infty)=1\,\,\forall n\in\mathbb N$
I wanted to combine the two somehow to show the result I require. Any hints?
I heuristically know the two statements I have already proved imply the condition I want to prove, but I was looking for a mathematical solution
The standard method to compute the probability of reaching some state is to write done a system of linear equation. Fix $n$ and for any $k \in \{0,1,2 \ldots \}$ write $h_k = \mathbb{P}_k (T_n < \infty)$. Markov property allows you write the following for any $k \in \{0,1,2 \ldots \}$
$$ h_k = (1-p)h_{0} + p h_{k+1}.$$
Now if you already proved that $h_0 = 1$ (as you claim you have), taking $k=0$ in the equation above gives you $h_1 = 1$. By induction you get that $h_n = 1$ as well, i.e. the result you were looking for.