Ex: Proposition: Prove that $\forall i\in\Omega,\exists j\in\Omega$ such that $i\to j$.
I know that a state $i$ leads to a state $j$, if $n\geqslant 1:P^{(n)}(i,j)>0$. $P$ stands for stochastic matrix or Markov matrix. However in the exercise there is nothing explicit about the connection of $i$ and $j$. I f I used the conditional probability I would get $P(X_1=j|X_0=i)=\frac{P(X_1=j,X_0=i)}{P(x=i)}$ If intersection is zero then $P(X_1=j|X_0=i)=0$ so $i$ would not lead to $j$ which contradicts the proposition I intended to prove.
Question:
How do I prove the proposition?
You should be more careful about the quantifiers $\forall$ and $\exists$.
Let $i \in \Omega$. We know that $\sum_{j\in\Omega} P(i,j) = 1$ by definition of a stochastic matrix, hence there exists $j \in \Omega$ such that $P(i,j) > 0$ (otherwise the sum would be $0$).