 Club of closure points for countable set-valued functions

For a regular uncountable cardinal $\lambda$ and $B:\lambda \to [\lambda {]}^{\omega }$, the limit ordinals closed under all earlier values of $B$ form a club set. Countably iterate bounds for ${\bigcup }_{\alpha <\gamma }B(\alpha )$ to obtain unboundedly many closure points. Regularity keeps each bound and the countable supremum below $\lambda$. Limits of closure points remain closure points.