Notice a potential bound such as $f_{L}\geq c$ can be characterized by the positivity of the measure $g\mapsto\int_{L}(f_{L}-c)g\text{Re}\Omega$. This characterization is robust under current convergence, so

$$\sup_{L_{i}}f_{L_{i}}\leq\limsup_{j}\sup_{L_{i}^{(j)}}f_{L_{i}^{(j)}},\quad\inf_{L_{i}}f_{L_{i}}\geq\liminf_{j}\inf_{L_{i}^{(j)}}f_{L_{i}^{(j)}},$$

hence the potential clustering bounds pass to the limit. ∎