I have a question to the following proof:
I do not understand the line
$$ \textbf{P}_{x}\{\tau_{y}^{+} > kr\} \leq (1-\varepsilon)\textbf{P}_{x}\{\tau_{y}^{+} > (k-1)r\} \tag{1.17}. $$
I think the claim in the text is that $$ \textbf{P}_{x}\{\tau_{y}^{+} \in \{t,t+1, \ldots, t+r\}\} \geq \varepsilon. \tag{$\star$} $$
This would imply
$$ \textbf{P}_{x}\{\tau_{y}^{+} \in \{(k-1)r,(k-1)r+1, \ldots, kr\}\} \geq \varepsilon $$ and $$ \textbf{P}_{x}\{\tau_{y}^{+} \notin \{(k-1)r,(k-1)r+1, \ldots, kr\}\} \leq 1 - \varepsilon $$ and
$$ \textbf{P}_{x}\{\tau_{y}^{+} > kr\} = \textbf{P}_{x}\{\tau_{y}^{+} > kr \mid \tau_{y}^{+} > (k-1)r\} \textbf{P}_{x}\{\tau_{y}^{+} > (k-1)r\} \\ \leq \textbf{P}_{x}\{\tau_{y}^{+} \notin \{(k-1)r,(k-1)r+1, \ldots, kr\}\} \textbf{P}_{x}\{\tau_{y}^{+} > (k-1)r\} $$ because of monotonicity ($\{\tau_{y}^{+} > kr \mid \tau_{y}^{+} > (k-1)r\} \subseteq \{\tau_{y}^{+} \notin \{(k-1)r,(k-1)r+1, \ldots, kr\}\}$).
If these thoughts are correct, my question reduces to the verification of $(\star)$. I think that this follows from the (strong?) Markov property, but I am not sure how to apply.
The text is correct. Your statement (*) is not valid.
It is true that with probility $\ge \epsilon$ the chain will visit state $y$ in the interval $\{t,t+1,...,t+r\}$, but not necessarily for the first time.
So you need to condition your calculation according to $P(\tau_y^+ \lt t)$