= Compact-sublevel convergence of the residual method
Suppose $J$ is nonnegative and <sequentially lower semicontinuous>, and its sublevel sets are strongly sequentially compact. For a solvable exact equation $Au=f$, every sequence of residual-method minimizers with $\delta\to0$ has a strongly convergent subsequence, and every limit is a <J-minimizing solution>. Indeed, comparison with an exact penalty minimizer bounds all reconstructions in one compact sublevel set; the residual bound gives $\|Au_\delta-f\|\leq2\delta$. Continuity of $A$ and lower semicontinuity of $J$ identify each subsequential limit as an exact minimizer. Thus the <distance to a set> of exact penalty minimizers tends to zero. Uniqueness of the exact minimizer upgrades this to full <strong convergence>.
Uniqueness cannot be omitted: for $A=0$, $f_\delta=f=0$, and $J(u)=\max\{|u|-1,0\}$ on $\mathbb R$, the sublevel sets are compact intervals but the minimizers $u_{1/k}=(-1)^k$ do not converge.
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