Understanding the triangle inequality of the essential supremum norm.(Royden 4th edition on pg.138)

Question

The proof is given below:

My question is:

1-it is not clear for me what is the idea they used to prove that $\| f\|_{\infty}$ is an essential upper bound for $f$ and what is the $E_{n}$ used in that case. could anyone explains this for me, please?

Answer 1

$\|f\|_{\infty}$ is defined as the infimum of all positive numbers $M$ such that $|f| \leq M$ almost everywhere. Since $\|f\|_{\infty}+\frac 1 n >\|f\|_{\infty}$ it follows that there exits $M$ such that $|f| \leq M$ almost everywhere and $\|f\|_{\infty}+\frac 1 n>M$. Take $E_n=\{x\in E: |f| > M\}$. This $E_n$ has the stated properties.