 Geometric correction for approximate surjectivity

Let $T:E\to F$ be a bounded linear operator, $E$ a Banach space, and $F$ a normed vector space. Suppose each residual $r$ has an approximate preimage of norm at most $R\Vert r\Vert$ leaving error at most $k\Vert r\Vert$, where $0<k<1$. Correct successive residuals to get vectors $x_j$ with $\Vert x_j\Vert \le R{k}^j\Vert y\Vert$. Their absolutely convergent sum in $E$ maps exactly to $y$, and its norm is at most $R\Vert y\Vert /(1-k)$. Completeness of $F$ is not needed for this argument.