= Solution
Let $u^\dagger=A^\dagger f$. Linearity and the <triangle inequality> give the <noise-bias decomposition for linear regularization>
$$
\|R_{\alpha(\delta)}f_\delta-u^\dagger\|
\leq\|R_{\alpha(\delta)}(f_\delta-f)\|
+\|R_{\alpha(\delta)}f-u^\dagger\|
\leq\delta\|R_{\alpha(\delta)}\|
+\|R_{\alpha(\delta)}f-A^\dagger f\|.
$$
The first term tends to zero by the assumed noise-amplification bound. The second tends to zero because $\alpha(\delta)\to0$ and $R_\alpha$ is a <regularization of an inverse problem>. The right side is independent of the particular noisy datum within its allowed ball. Its convergence therefore proves the uniform noisy-data convergence required of a <convergent regularization of an inverse problem>.
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