= Solution
Bounded means finite diameter; sequential compactness means every <sequence> has a convergent subsequence. A failure of uniform continuity supplies two close <sequences> whose image distances stay apart; a convergent subsequence contradicts continuity. Into a discrete metric, continuity is local constancy, and compactness upgrades the radii uniformly. The Cantor set is closed in compact $[0,1]$, hence sequentially compact. Uniform local constancy gives a positive separation scale, so finitely many initial Cantor-code (binary) digits determine $f$. Removing zero destroys compactness: the hinted <function> reading the digit after the first $1$ is continuous at every remaining <sequence> but depends on arbitrarily late digits.
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