= Solution
\b[Weakly inaccessible cardinals are unbounded below the given cardinal.] In fact, they form a <stationary set> there. Let $\kappa$ be the <weakly Mahlo cardinal>, so it is a regular uncountable <limit cardinal> and the set $R=\{\alpha<\kappa:\alpha=\operatorname{cf}(\alpha)\}$ is stationary.
The set $C$ of uncountable <limit cardinals> below $\kappa$ is a <club set>. For unboundedness, above any starting point choose a strictly increasing countable sequence of <cardinal numbers> below $\kappa$; its supremum remains below $\kappa$ by regularity and is an uncountable <limit cardinal>. For closure, a limit of such limit cardinals is again a <limit cardinal>.
Every $\alpha\in R\cap C$ is an uncountable <regular cardinal> and a <limit cardinal>, hence a <weakly inaccessible cardinal>. The intersection of a <stationary set> with a <club set> is stationary, because its intersection with any further club is nonempty. Therefore
$$
\boxed{\{\lambda<\kappa:\lambda\text{ is weakly inaccessible}\}\text{ is stationary, hence unbounded}.}
$$
This proves the stronger form of the requested conclusion.
Back to article page
