In the proof of the "Law of Large Numbers for the Brownian Motion" the following claim is made:
Let $(B_t)_{t\in \mathbb R}$ be (a continuous version of) a Brownian motion. Fix some time $t>0$. If we define $$X_n^\ast := \max_{0 \leq i \leq 2^n} \lvert B_{t+i2^{-n}} - B_t \rvert, \quad n\in \mathbb N,$$ then $$X_n^\ast \stackrel{n\to \infty}{\longrightarrow} \sup_{0 \leq s \leq 1} \lvert B_{t+s} - B_t \rvert \quad \mathbb P\text{-a.s}.$$ by the continuity of the paths.
I tried to prove this assertion which apparently was obvious to the author, but I had some issues in the end. For $n\in \mathbb N, 0 \leq i \leq 2^n$, the inequality $$\lvert B_{t+i2^{-n}} - B_t \rvert \leq \sup_{0 \leq s \leq 1} \lvert B_{t+s} - B_t \rvert$$ proves $X_n^\ast \leq \sup_{0 \leq s \leq 1} \lvert B_{t+s} - B_t \rvert$. Now aiming to apply the sandwich theorem, I tried to prove a converse inequality with some tolerance. Let $s\in [0,1]$. By continuity of the Brownian paths, choose $\delta = \delta(n) > 0$ such that $\lvert s-x \rvert < \delta$ implies $\lvert B_s - B_x \rvert < \frac{1}{n}$. By the density of dyadic numbers in $[0,1]$, pick $m\in \mathbb N$ and $0 \leq i \leq 2^m$ such that $\lvert s -i2^{-m} \rvert < \delta$. It follows that $$\lvert B_{t+s} - B_t \rvert \leq \lvert B_{t+s} - B_{t+i2^{-m}} \vert + \lvert B_{t+i2^{-m}} - B_t \rvert < \frac{1}{n} + X_m^\ast.$$ Here is where I am stuck. My problem is that I can not relate $n$ and $m$. The goal is to have only $n$ on the right side and then send it to $\infty$. Any hints?
You are almost there, and you don't need to "relate" $n$ and $m$.
Just note that the inequality doesn't just hold for one specific $m$; it holds for all sufficiently large $m$. So in fact, for each fixed $n$ and $s$, you can pass to the limit and conclude $$|B_{t+s} - B_t| \le \frac{1}{n} + \liminf_{m \to \infty} X^*_m.$$ Now let $n \to \infty$: $$|B_{t+s} - B_t| \le \liminf_{m \to \infty} X^*_m$$ and finally take the supremum over $s$:
$$\sup_{0 \le s \le 1} |B_{t+s} - B_t| \le \liminf_{m \to \infty} X^*_m.$$
Your previous work showed that $\limsup_{m \to \infty} X^*_m \le \sup_{0 \le s \le 1} |B_{t+s} - B_t|$, so you are done.