A question on the proof of the inequality $ \sum_{k} |I_{k}| \leq |I| $, where $ I $ and the $ I_{k} $’s are intervals.

Question

Theorem. If $ \displaystyle \bigcup_{k} I_{k} \subseteq I $ and the $ I_{k} $’s are disjoint, then $ \displaystyle \sum_{k} |I_{k}| \leq |I| $.

I can understand the proof for the finite case, but when it passes to the infinite case, I am quite lost. The text claims that “if there are infinitely many intervals, then there exist finite subcollections that satisfy finite additivity”.

My question is: How do we know that there exist finite subcollections from a collection of infinitely many intervals?

Answer 1

For any set $X$ and integer $n$ you can form the subset of the power set of X $$X_n:= \{A\subset X: \#A = n\}$$ which will be not empty (where # denotes the cardinality). If now $X$ is your collection of intervals then this way you get the finite subcollections.