Let X be a right-continuous Feller-Dynkin process and define the stopping time $$\nu_{r}=\inf\{t\geq 0\mid ||X_{t}-X_{0}||\geq r\}$$
Let $B_{x}(\epsilon)=\{y\mid ||y-x|| \leq \epsilon\}$, for $x$ not absorbing we then have for the transition function $$P_{t}(x,B_{x}(\epsilon))<p<1$$ for some $t,\epsilon>0$
For $\hat{p}\in(p,1)$ it follows that for $y\in B_{x}(r)$ for some $r\in(0,\epsilon)$ that $$P_{t}(y,B_{x}(r))\leq \hat{p}$$
My question is, how can I show that $P_{x}(\nu_{r}>nt)\leq \hat{p}^{n}$ holds using the Markov property of X??