Understanding some points of "Lemma 10" in Royden "Real Analysis" fourth edition.(2)

Question

The lemma and its proof are given below:

My question is:

1-In the second line from below: Why we are guaranteeing that we can choose an index $N$ for which $m(E_{N}) > m(E) - \epsilon.$? could anyone help me in answering this question, please?

2-Also, it is not so clear for me why $E_{n}$ are ascending, could anyone explain this for me, please?

Answer 1

1. Because $m(E)=\lim_{n\to\infty} m(E_n)$ (and $m(E)$ is just a real number)
2. Because if $n_1\le n_2$ and $x$ satisfies $\lvert f(x)-f_k(x)\rvert<\eta$ for all $k\ge n_1$, then a fortiori it satisfies $\lvert f(x)-f_h(x)\rvert<\eta$ for all $h\ge n_2$.