The following is part of my textbook about proving the expectation of Markov chain first hitting time is finite. I understand everything except for the circled inequality $\sum\limits_{t \ge 0} {P\{ r_y^ + > t\} } \le \sum\limits_{k \ge 0} {rP\{ r_y^ + > kr\} } $.
I know $P\{ r_y^ + > t\}$ is decreasing because event $\{ r_y^ + > t + 1\} \subseteq \{ r_y^ + > t\} $, but I just have difficulty understanding how $t$ is replaced by $kr$ and why the inequality holds. Anyone can help explain? Thank you!
For $kr\le t\le (k+1)r$, $P(\tau > t)\le P(\tau>kr)$, so$\sum_{t\ge0}P(\tau>t)=\sum_{k\ge0}\sum_{i=0}^{r-1}P(\tau>kr+i)\le\sum_{k\ge0}rP(\tau>kr).$