A Closed Interval Covered by Open Intervals Is No Longer than Their Total Length

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\mathbb{R}$, its order and its arithmetic be as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ii), the intervals $[p,q]$ and $(a,b)$ as in Intervals of the Real Line (ii), $[0,\infty]$ and its order $\le$ as in The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii), and its sum $+$ as in Addition and Multiplication on [0,∞] (i). Sums over finite sets of maps into $[0,\infty]$ are as in Sums and Products over a Finite Set and over an Interval (i), Sums and Products over a Finite Set and over an Interval (ii), Sums and Products over a Finite Set and over an Interval (iii) and Sums and Products over a Finite Set and over an Interval (v), for $+$, which is associative and commutative with neutral element $0$ by Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (i), and sums of sequences in $[0,\infty]$ are as in Sum of a Sequence in [0,∞] (ii). Let $(a_{m})$ and $(b_{m})$ be sequences in $\mathbb{R}$ with $a_{m}\le b_{m}$ for every $m\in\mathbb{N}$, so that $b_{m}-a_{m}\in[0,\infty]$, and let $p,q\in\mathbb{R}$ with $p\le q$.

(i) Let $F$ be a finite subset of $\mathbb{N}$ such that every $x\in[p,q]$ lies in $(a_{m},b_{m})$ for some $m\in F$. Then

$$q-p\le\sum_{m\in F}(b_{m}-a_{m}).$$

(ii) If every $x\in[p,q]$ lies in $(a_{m},b_{m})$ for some $m\in\mathbb{N}$, then

$$q-p\le\sum_{m\in\mathbb{N}}(b_{m}-a_{m}).$$