= Convergence on a dense subspace under a uniform operator bound
{title2=$\sup_h\|A_h\|<\infty,\quad A_hd\text{ converges on dense }D\ \Longrightarrow\ A_hx\text{ converges for every }x$}
Let $A_h:X\to Z$, $0<h<1$, be <bounded linear operators> from a <normed space> into a <Banach space>, with $M=\sup_h\|A_h\|<\infty$. If $A_hd$ converges as $h\to0^+$ for each element of a <dense subspace> $D$, then $A_hx$ converges for every $x\in X$. The limit defines a <bounded linear operator> $A$ with $\|A\|\leq M$.
For $d\in D$, the <triangle inequality> gives
$$
\|A_hx-A_kx\|\leq2M\|x-d\|+\|A_hd-A_kd\|.
$$
Approximate $x$ by $d$ and use the <Cauchy criterion for a function limit> on $D$. The <completeness> of $Z$ supplies the limit; passing to the limit preserves <linearity> and the bound. When every limit on $D$ is zero, the inequality $\|A_hx\|\leq M\|x-d\|+\|A_hd\|$ shows $A=0$.
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