Can anybody help me to show the first inequality in this proof? It is about standard Brownian Motion. So B(t) is a Brownian Motion and I know the rescaling and shifting properties of Brownian Motions but I don't know how to handle the supremum and how to get the $2^n m^2$ out of the $\mathbb{P}$?
The second inequality is clear.
Would be great if somebody could give me a hint!
Thanks for your help.
This is just an application of the union bound: $$P\left(\sup \limits_{i = 1}^n X_i > c\right) = P\left(\bigcup \limits_{i = 1}^n \{X_i > c\}\right) \le \sum \limits_{i = 1}^n P(X_i > c) \le n \sup \limits_{i = 1}^n P(X_i > c)$$