If $\kappa$ is the $\alpha<\kappa$-th inaccessible cardinal, then the regular cardinals below $\kappa$ form a non-stationary set

Question

How can I prove that if $\kappa$ is the $\alpha<\kappa$-th inaccessible cardinal, then the set of all regular cardinals below $\kappa$ is nonstationary? Is it because the set $\{x: x\text{ is a singular strong limit cardinal smaller than }\kappa\}$ is a club in $\kappa$?

Thank you so much for your answers.

Answer 1

Well, it's easier to do it by contrapositive.

Since $\kappa$ is strongly inaccessible, the set of strong limit cardinals is a club below $\kappa$. If the regular cardinals form a stationary set, the intersection with the aforementioned club is stationary. So there is a stationary set of inaccessible cardinals.

In particular, there are unboundedly many inaccessible cardinals below $\kappa$, so it has to be the $\kappa$th inaccessible.