= Solution
A finite right <derivative> at zero would make the difference quotients $B_{1/n}/(1/n)$ eventually bounded. Therefore that event is contained in
$$
\bigcup_{K,N\geq1}\left\{|B_{1/n}|/(1/n)\leq K\text{ for every }n\geq N\right\}.
$$
For fixed $K,N$, its probability is at most $\mathbb P(|B_{1/n}|\leq K/n)$ for each $n\geq N$. By the <normal distribution> of the Brownian increment,
$$
\mathbb P(|B_{1/n}|\leq K/n)=2\Phi(K/\sqrt n)-1\longrightarrow0.
$$
Thus every event in this countable union has probability zero. \b[With probability one, <Brownian motion> has no finite right derivative at zero], which is the appropriate derivative for its time domain. No independence of the quotients is assumed or needed. This proves the required endpoint case of <nowhere differentiability of Brownian motion>.
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