= Solution
The $2^n$ increments are independent <normal random variables>, each distributed as $2^{-n/2}B_1$. Their squared means are $2^{-n}$ and, using the permitted <variance>, their squared variances are $2\cdot2^{-2n}$. <Independence> therefore gives
$$
\mathbb E[B]_n=2^n2^{-n}=1,
\qquad \operatorname{var}([B]_n)=2^n\,2\cdot2^{-2n}=2^{1-n}.
$$
Consequently \b[the dyadic quadratic variation converges to one]:
$$
\boxed{\mathbb E\bigl([B]_n-1\bigr)^2=2^{1-n}\longrightarrow0,
\qquad [B]_n\longrightarrow1\text{ in }L^2.}
$$
This is <dyadic quadratic variation of Brownian motion> on the time interval of length one. The calculation requires <independence> only within each partition; increments from different resolutions need not be independent.
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