Let $(X_{n})_{n \in \mathbb N}$ be a Markov Chain
How can I show $\sum\limits_{y \in E}\sum\limits_{n \geq 0}P^{x}(\tau_{x}>n,X_{n}=y)\Pi(y,z)=\sum\limits_{n \geq 0}\sum\limits_{y \in E}P^{x}(\tau_{x}>n,X_{n}=y,X_{n+1}=z)$ using the Markov Property. Note that $\tau_{x}:=\inf\{n \in \mathbb N: X_{n}=x\}$ and $\Pi$ represents the transition matrix on our state space $E$.
My ideas:
$P^{x}(\tau_{x}>n,X_{n}=y)\Pi(y,z)=P^{x}(\tau_{x}>n,X_{n}=y,X_{n+1}=z)$ because the event that we transition from $y$ to $x$ of the interval $n+1$ is independent of the event $\{\tau_{x} > n\}$.
And now I cannot simply change the summations, because one sum $\sum\limits_{n \geq 0}$ sums up to infinity, correct? So then how can I justify switching the summations?
You can still switch the summations as the summands are all non-negative. This is Tonelli's theorem. Here is a reference.