= Solution
Construct one infinite <binary sequence> $u$ by concatenating every binary word of length one, then every word of length two, then every word of length three, and so on, in <lexicographic order> within each length. Every finite binary word occurs somewhere in $u$.
Given any target <sequence> $s$ and any $\eta>0$, choose $N$ such that $2^{-N}<\eta$. The word $s_1\cdots s_N$ occurs in $u$, beginning after some $k$ symbols. Thus $\sigma^ku$ agrees with $s$ in its first $N$ positions, and the prescribed <metric> gives
$$
d(\sigma^ku,s)\leq\sum_{i>N}2^{-i}=2^{-N}<\eta.
$$
Every open <open ball> about every point therefore meets the <forward orbit> of $u$. \b[This <forward orbit> is <dense> in $\Sigma$.] The argument identifies finite-prefix <cylinder sets> as arbitrarily small neighborhoods in this <metric>; no binary-expansion convention for real numbers is needed.
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