How to show this inequality holds, when proving self-similar processes are non-differentiable? In the book Elementary Stochastic Calculus by T. Mikosch (1998), there is a result which shows that Brownian motion is a self-similar process and therefore it is nowhere differentiable. In the proof of this result, there is an inequality:
$$\lim\limits_{n\to\infty}P(\mathrm{sup}_{0\leq s \leq t_n}|\frac{X_s}{s}|>x )\geq {\lim\limits \text { sup}}_{n\to\infty}P(|\frac{X_{t_n}}{t_n}|>x)$$
where $(X_t)$ is a self-similar process. I have two questions: (1) Can we interchange $P()$ and $\mathrm{sup}$ operation? (2) how to show the inequality $\geq$ here? Thank you.

Look at the section Borel-Cantelli Lemmas in Durret's Book (Probability Theory and Examples) an application of Fatou's Lemma ensure that $P(\limsup_n An)\geq \limsup_n P(A_n)$. So if you write $A_n$ for the set if LHS of the inequality and $B_n$ for the another then $$\lim_n P(A_n)=P(\lim_n A_n)=P(\limsup_n A_n)\geq P(\limsup_n B_n)\geq \limsup_n P(B_n)$$
