Let $X$ be a right-continuous Feller Dynkin process. For $r>0$ we define the $\{\mathcal{F}_t\}_t$ stopping time (which is called escape time) $$\eta_r=\inf\{t\geq 0: \|X_t -X_0\|\geq r\}$$ We have that $P_t(y,B_y(r))\leq p$, where $B_y(r)$ is the closed ball of radius $r$ around the point $y$.
Now we need to prove that $P_x(\eta_r>nt)\leq p^n$, which can be shown by using the Markov property.
However, I can not figure out the details. Can anyone give me a detailed proof, since it is intuitively very clear.
By definition,
$$\begin{align*} \mathbb{P}^x(\eta_r > nt) &= \mathbb{P}^x \left( \sup_{s \leq nt} |X_s-X_0| < r \right) \\ &= \mathbb{E}^x \left( 1_{\{\sup_{s \leq t(n-1)} |X_s-X_0| < r\}} \cdot 1_{\{\sup_{t (n-1) \leq s \leq nt} |X_s-X_0| < r \}} \right). \end{align*}$$
Conditioning on $\mathcal{F}_{(n-1)t}$ yields, by the Markov property,
$$\begin{align*}\mathbb{P}^x(\eta_r > nt)&= \mathbb{E}^x \bigg[1_{\{\sup_{s \leq t(n-1)} |X_s-X_0| < r\}} \mathbb{P}^{X_{t (n-1)}} \left( \sup_{s \leq t} |X_s-X_0|<r \right) \bigg] \\ &= \mathbb{E}^x \bigg( 1_{\{\sup_{s \leq t(n-1)} |X_s-X_0| < r\}} \underbrace{P_t(y,B(y,r)) \bigg|_{y=X_{(n-1)t}}}_{\leq p} \bigg) \\ &\leq p \, \mathbb{P}^x \left( \sup_{s \leq t (n-1)} |X_s-X_0| < r \right) \end{align*}$$
Iterating the procedure, finishes the proof.