For each finite n, let μn+1 = 2μn which is less than κ because it is a strong limit cardinal. Then their limit is a strong limit cardinal and is less than κ by its regularity. The limits of uncountable strong limit cardinals are also uncountable strong limit cardinals. So the set of them is club in κ. Intersect that club set with the stationary set of weakly inaccessible cardinals less than κ to get a stationary set of strongly inaccessible cardinals less than κ. Example: showing that Mahlo cardinals κ are κ-inaccessible (hyper-inaccessible) The term "hyper-inaccessible" is ambiguous. In this section, a cardinal κ is called hyper-inaccessible if it is κ-inaccessible (as opposed to the more common meaning of 1-inaccessible). Suppose κ is Mahlo. We proceed by transfinite induction on α to show that κ is α-inaccessible for any α ≤ κ. Since κ is Mahlo, κ is inaccessible; and thus 0-inaccessible, which is the same thing. If κ is α-inaccessible, then there are β-inaccessibles (for β < α) arbitrarily close to κ. Consider the set of simultaneous limits of such β-inaccessibles larger than some threshold but less than κ. It is unbounded in κ (imagine rotating through β-inaccessibles for β < α ω-times choosing a larger cardinal each time, then take the limit which is less than κ by regularity (this is what fails if α ≥ κ)). It is closed, so it is club in κ. So, by κ's Mahlo-ness, it contains an inaccessible. That inaccessible is actually an α-inaccessible. So κ is α+1-inaccessible. If λ ≤ κ is a limit ordinal and κ is α-inaccessible for all α < λ, then every β < λ is also less than α for some α < λ. So this case is trivial. In particular, κ is κ-inaccessible and thus hyper-inaccessible. To show that κ is a limit of hyper-inaccessibles and thus 1-hyper-inaccessible, we need to show that the diagonal set of cardinals μ < κ which are α-inaccessible for every α < μ is club in κ. Choose a 0-inaccessible above the threshold, call it α0. Then pick an α0-inaccessible, call it α1. Keep repeating this and taking limits at limits until you reach a fixed point, call it μ. Then μ has the required property (being a simultaneous limit of α-inaccessibles for all α < μ) and is less than κ by regularity. Limits of such cardinals