= Successive approximation for quotient operators
{title2=$\sum_{n\geq0}\delta^n=(1-\delta)^{-1}$}
Let $T:X\to Y$ be a <bounded linear operator> with $X$ a <Banach space>. Suppose every $\|y\|<1$ has an approximation $Tx$ with $\|x\|<1$ and error less than $\delta\in(0,1)$. Repeatedly approximate the scaled residual to construct $x_n$ with $\|x_n\|<\delta^n$ and residual less than $\delta^{n+1}$. Completeness sums the <series>, and continuity gives $T\sum x_n=y$. Since $\|\sum x_n\|<(1-\delta)^{-1}$, scaling proves $B_Y(0,1-\delta)\subseteq T(B_X(0,1))$. Hence $T$ is a <quotient operator>, even without assuming completeness of $Y$.
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