set theory - Some consequences of internally approachable structures

I just read for the first time the definition of an internally approachable set, which says:

A set $N$ is internally approachable (I.A.) of length $mu$ iff there is a sequence $(N_{alpha} : alpha < mu)$ for which the following holds: $N=bigcup_{alpha< mu} N_{alpha}$ and for all $beta < mu$ $( N_{alpha} : alpha < beta ) in N$.

Now if $N prec (H(theta), in, < )$ is I.A. of length $mu$. Is it true that

(a) If $alpha < mu$ then $alpha in N$

(b) If $alpha < mu$ then $N_{alpha} in N$ ?

This is trivial if $N$ is transitive, and I'm quite sure that both (a) and (b) hold but I need a good argument.