How does one prove outer measure is finite for $A \in \mathfrak{M}_F(\mu)$ if there exist elementary sets $A_n \to A$?

Question

I was trying to prove for $A \in \mathfrak{M}_F(\mu)$ that if $A_n \to A$ with $A_n$ elementary sets, then $\mu^*(A)$ is finite. (All notations and definitions consistent with Rudin's Principles of Mathematical Analysis.) I get that $(\mu^*(A_n) - \mu^*(A)) \to 0$, and for each $n$ that $\mu^*(A_n)$ is finite, but I cannot understand how that implies $\mu^*(A) < \infty$. I would appreciate any help.

Answer 1

$\mu^*(A_n) - \mu^*(A) >-1$ for some $n$ (in fact for all $n$ sufficiently large). Hence, $\mu^*(A)<\mu_n^*(A)+1<\infty$.