My question concerns regarding the following lemma:-
"Lemma:- $[a_1,b_1)$,$[a_2,b_2)$ ... be a partition of an interval $[a,b)$, that is the intervals $[a_1,b_1)$,$[a_2,b_2)$ ... are disjoint and $$\bigcup\limits_{n=1}^\infty [a_n,b_n)=[a,b) $$ Then $$\Sigma_{n=1}^{\infty}(b_n - a_n)=b-a$$
The proof is here. My doubts are as follows:-
Why can't we directly conclude it as follows:-if the semi open sets are of the form$[x_n,x_{n+1})$, then the sum $\Sigma_{n=1}^{\infty}(x_{n+1}-x_{n})=b-a$ as seen from this document.
2.Also in the proof it is said that we assume a non-decreasing sequence of elements of S. But why is this so? As per the proof upto this part we only know that there is atleast one element but how do we know there is more than one and also that they form a converging sequence of elements to s?(the supremum).
in the statement you are considering here, it is not given that $a_{i+1}=b_i$ for any $i$. So the reasoning of your alternative proof cannot be applied directly. If you had a finite partition you could rearrange, but with countably many this is not so easy.
if you know a nonempty set has a lub, you can always find a nondecreasing sequence in that set converging to the lub, almost by definition (you should check that). If the set has only one element, then the sequence is constant, but so what?