= Solution
Let $w=K^*Kv\in\partial J(u^\dagger)$ from the <normal-operator source condition>, and set $e=u-u^\dagger$. Since $u^\dagger$ is a <least-squares solution>, the <normal equation for a linear inverse problem> gives $K^*(Ku^\dagger-f)=0$. Expand the objective difference and use the <Bregman distance> definition:
$$
\begin{aligned}
\mathcal F_\alpha(u)-\mathcal F_\alpha(u^\dagger)
&=\frac12\|Ke\|^2+\alpha[J(u)-J(u^\dagger)]\\
&=\frac12\|Ke\|^2+\alpha\langle Kv,Ke\rangle+\alpha D_J^w(u,u^\dagger)\\
&=\frac12\|K(e+\alpha v)\|^2+\alpha D_J^w(u,u^\dagger)-\frac{\alpha^2}2\|Kv\|^2.
\end{aligned}
$$
The cross term with the residual of the <least-squares solution> vanishes by the normal equation; exact data are not required. Compare the minimizing $u_\alpha$ with $z=u^\dagger-\alpha v$. At $z$, the completed square is zero. Dividing the resulting inequality by $\alpha$ proves the stronger <shifted-comparator Bregman bound>
$$
\boxed{D_J^w(u_\alpha,u^\dagger)+\frac1{2\alpha}\|K(u_\alpha-u^\dagger+\alpha v)\|^2\leq D_J^w(u^\dagger-\alpha v,u^\dagger),\qquad w=K^*Kv.}
$$
Dropping the nonnegative squared term gives the requested estimate. If the comparator lies outside the effective domain of $J$, its distance is infinite and the inequality is valid but gives no finite error bound.
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