In Discrete Markov Chain we can define the First-Passage Probabilities:
$$f_{ij}^{(m)}=P\{X_{n+1}\neq j,\dots,X_{n+m-1}\neq j, X_{n+m}=j|X_n=i\}$$
What I would like to know is how to prove the following equation:
$$\sum_{m=1}^\infty f_{ij}^{(m)}\le1$$
And why we do not have the following equation:
$$\sum_{m=1}^\infty p_{ij}^{(m)}\le1$$
where $p$ is the transition probability. In brief, what I would like to know is the difference between these two from the view of probability.