I've studied a proof from the book An Introduction to Stochastic Differential Equations concerning the nowhere differentiability of the Brownian motion and I'm stuck at the following proof:
In the second point, I don't understand the third inequality
$$\liminf_{n\to\infty} \sum_{i=1}^n P(A_{M,n}^i) \leq \liminf_{n\to\infty} n\cdot P(|W(1/n)|\leq Mn^{-\gamma})^N.$$
Which theorem in the probability theory was applied in this inequality?
In fact, the inequality in question follows from the following equality: $$ \mathbb P(A^i_{M,n})=\mathbb P(|W(1/n)|\leq Mn^{-\gamma})^N. $$ It is a direct consequence of the independence of increments property of the Brownian motion combined with stationarity of increments.
The former property implies that the events $$ \left\{\omega\colon \left|W\bigl(\frac{j+1}{n}\bigr)-W\bigl(\frac{j}{n}\bigr)\right|\leq Mn^{-\gamma}\right\},\qquad j=i,\ldots,i+N-1 $$ are independent. The latter means that $$ W\bigl(\frac{j+1}{n}\bigr)-W\bigl(\frac{j}{n}\bigr) $$ all have the same distribution, regardless of $j$, and equal to the distribution that results when $j=0$.
Combining these two properties yields the equality I mentioned at the start.