So I've been trying to teach myself some set theory and I've come across some exercises in Just and Weese's Discovering Modern Set Theory. To whit:
Definition 20: A cardinal $\kappa$ is called weakly inaccessible if $\kappa$ is an uncountable regular limit cardinal.
Exercise 27 (PG): Show that if $\alpha$ is a weakly inaccessible cardinal, then $\alpha=\aleph_\alpha$.
Exercise 28 (R): Show that the smallest ordinal $\alpha$ such that $\alpha=\aleph_\alpha$ is not a weakly inaccessible cardinal.
For the first I've proved by induction that $\alpha\subseteq\aleph_\alpha$ for all ordinals $\alpha$, obviously I have to use the weakly inaccessible part to go the other way, but I've no idea where to start. Weakly inaccessible gives that every function with cofinal range in $\alpha$ has domain at least $\alpha$ but that is saying that things are big, where as I need that $\aleph_\alpha$ is small.
The second is beyond me, I considered trying to show that ZFC proves the existence of such an ordinal $\alpha$, and then appeal to the fact that the existence of a weakly inaccessible cardinal is independent of ZFC, however this result is yet to be obtained in the text and so I imagine this can't be what is wanted.