= Solution
Suppose each $R_\alpha:\mathcal V\to\mathcal U$ is a <bounded linear operator> and $R_\alpha f\to K^\dagger f$ for every $f\in\mathcal D(K^\dagger)$ as $\alpha\downarrow0$. A standard sufficient <a priori regularization parameter choice> satisfies
$$
\boxed{\alpha(\delta)\longrightarrow0,\qquad
\delta\|R_{\alpha(\delta)}\|\longrightarrow0\quad(\delta\downarrow0).}
$$
For every datum with $\|f^\delta-f\|\leq\delta$, the <noise-bias decomposition for linear regularization> gives
$$
\|R_{\alpha(\delta)}f^\delta-K^\dagger f\|
\leq\delta\|R_{\alpha(\delta)}\|
+\|R_{\alpha(\delta)}f-K^\dagger f\|\longrightarrow0.
$$
Thus \b[the parameter rule gives a <convergent regularization of an inverse problem>], uniformly over data in the prescribed noise ball for each fixed admissible exact datum. For <Tikhonov regularization>, $\|R_\alpha\|\leq1/(2\sqrt\alpha)$, so $\alpha\to0$ and $\delta/\sqrt\alpha\to0$ suffice. The same sufficient noise scaling holds for <spectral cutoff regularization>.
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