The theorem and its proof (from Royden " Real Analysis " forth edition ) is shown below:
My questions are:
$1$-In case($1$), I do not understand why what is written leads to $$\lim_{n \rightarrow \infty} \psi_{n}(x) = f(x).$$ Could anyone explain this for me please?
$2$-In case($2$), why when $\lim_{n \rightarrow \infty} \phi_{n}(x) = f(x)$ this leads to that the sequence $\{ \psi_{n}\}$ converges to $f$ pointwise on $E.$Could anyone explain this for me please?
$3$-where is the proof of $|\phi_{n}| \leq |f|$ on $E$ for all $n$?
$4$-Why if we replace each $\phi_{n}$ with max $\{ \phi_{1}, ..., \phi_{n}\}$, we will have $\{\phi_{n}\}$ increasing?
Could anyone explain all the above points for me please?
Thanks!