On page 135 of Jech there is a step that I do not understand in the proof of "measurable implies Mahlo". We have already proved that "measurable implies strong limit". Jech writes
"As $\kappa$ is a strong limit, the set of all strong limit cardinals $\alpha < \kappa$ is closed unbounded...."
Surely every strong limit cardinal does not have a club set of strong limit cardinals below it (take the least strong limit cardinal, for example).
Am I missing something?