How to define $B_{\omega}$ where $B_{n+1}=\{\text{limits of functions in $B_n$\}}$ and $B_0=\{\text{continuous real functions}\}$

Question

This is Ex. $6.1.3$ in Stillwell's "Real Numbers," with the heading

Classes of Functions Obtained as Limits

I would appreciate help:

Defining $B_{\omega}$ where $B_{n+1}=\{\text{limits of functions in $B_n$\}}$ and $B_0=${continuous real functions}

It is supposed that each $B_{n+1}$ has members not in $B_n$.

(I really don't like to say that I don't know what to do and to not show any work, yet that is the case.)

Answer 1

I believe two (slightly) different conventions have been used for this. The one I prefer is $B_\omega=\bigcup_{n\in\mathbb N}B_n$. The other makes $B_\omega$ the set of limits of functions in $\bigcup_{n\in\mathbb N}B_n$ (which would be $B_{\omega+1}$ under my preferred convention).