 Dense open-unit-ball image criterion

Let $T:X\to X$ be a bounded linear operator on a Banach space, and let $B$ be the open unit ball. If $B\subseteq \overline{T(B)}$, then $B\subseteq T(B)$. Scale the density statement and correct a target by a geometrically decreasing sequence of approximate preimages. Completeness makes their sum converge inside $B$, and continuity of $T$ makes its image equal the target.