For a compact subset of $\mathbb{R}$, measure zero and content zero coincide

Given: A compact set $K\subseteq \mathbb{R}$ and a real $\epsilon >0$. Throughout, $\theta :=2^{-1}$.

$A$ is null when for every real $\eta >0$ there are sequences $(a_k)$, $(b_k)$ with $a_k\le b_k$, $A\subseteq {\bigcup }_k[a_k,b_k]$ and ${\sum }_{k<i}(b_k-a_k)\le \eta$ for every $i$; $A$ has content zero when the same holds with a finite list (Measure zero (a countable cover by intervals of total length below every $\epsilon$) and content zero (a finite such cover)).

A set of content zero is null (A set of content zero has measure zero).

$[c,d]$ has length $d-c\ge 0$ for $c\le d$; $(c,d)$ is the open interval with the same endpoints and is contained in $[c,d]$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

Every open interval is an open set (Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

$K$ is compact: from every family of open sets whose union contains $K$, either $K=\mathrm{\varnothing }$ and the empty subfamily covers it, or there are $m\in \mathbb{N}$ and members $U_0,\dots ,U_m$ of the family whose union contains $K$; compactness is equivalent to being closed and bounded (Open cover, subcover, compact subset of $\mathbb{R}$ (every open cover has a finite subcover), and sequentially compact subset, A subset of $\mathbb{R}$ is compact if and only if it is closed and bounded).

Powers and the geometric series: ${\theta }^0=1$, ${\theta }^{k+1}={\theta }^k\theta$, ${\theta }^k>0$, and ${\sum }_{k=0}^{\mathrm{\infty }}{\theta }^k=2$ for $\theta =2^{-1}$; a series of nonnegative terms has all its partial sums at most its sum (Integer powers $a^m$, For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges, Series, partial sums, convergence and the sum, divergence, and the tail series, A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum).

Finite sums: additivity, scaling, splitting and monotonicity in the terms (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

Every finite list of naturals has an upper bound in $\mathbb{N}$, by induction on its length and the totality of the order of $\mathbb{N}$ (The principle of mathematical induction, Trichotomy of the order on $\mathbb{N}$, Order on the natural numbers).

Ordered-field arithmetic: $0<1$, so $2>0$, $4>0$, $8>0$ and $t\cdot 8^{-1}>0$ for $t>0$; adding a constant and multiplying by a positive preserve an inequality (The multiplicative identity is positive, Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field, Complete ordered field (least-upper-bound property)). These order-arithmetic facts are stated by their sources for the strict order only; the nonstrict forms used below follow by adjoining the equality case, in which the two sides coincide.