I am trying to understand the proof of the below corollary from René Schilling's Brownian Motion. I have been able to follow the proof to get the inequality (19.38).
To prove the existence of the continuous modification, the author calls for the Kolmogorov-Chentsov Theorem, also given below. It says that the existence of a jointly continuous modification $(t,x)\mapsto X_t^x$ follows from the Kolmogorov-Chentsov theorem, Theorem 10.1, for the index set $\mathbb{R}^n \times [0,\infty) \subset \mathbb{R}^{n+1}$ and with $p=\alpha = 2(\beta+n+1).$
So from Theorem 10.1, we have $x=(x,s), y=(y,t)$ and the index set with dimension $n+1$.
Hence, we need to show that $C(|x-y|^{2(\beta+n+1)}+(1+|x|+|y|)^{2(\beta+n+1)}*|s-t|^{\beta+n+1}|(s,x)-(t,y)|^{n+1+\beta}) \le c|(x,s)-(y,t)|^{n+\beta+1}.$
How can we easily see that this inequality holds for some constant $c$?
