I have understood the overall proof but I have a confusion at the very beginning of the proof: Usually in the first case of a closed interval, say $[a,b]$, we find that for each $\varepsilon>0,$ $\exists$ an open interval $(a-\varepsilon,b+\varepsilon)$ which contains $[a,b]$. And then it is written that:
$m^*([a,b])\le \ell((a-\varepsilon,b+\varepsilon))=b-a+2\varepsilon.\tag1$
Herein lies my confusion. I mean, here $[a,b]\subseteq(a-\varepsilon,b+\varepsilon).$ So we could use the monotonicity property of outer measure, but then also we would have outer measure of that open interval on right in $(1).$ Instead, in $(1),$ the length of that open interval has been taken. Also, no open covering of $[a,b]$ has been taken so that we could use the sum of length of those open intervals on right in $(1).$
Basically, my question is: How could we justify the inequality $(1)$? Please shed some light about the same. Thanks in advance.