Proof of The Nondecreasing Envelope of a Truncated Modulus of Continuity

Each result cited is universally quantified over the data in its own statement. Two facts about suprema are used throughout, both contained in the definition of the least upper bound: for a nonempty $S\subseteq\mathbb{R}$ bounded above, every $s\in S$ satisfies $s\le\sup S$, and $\sup S\le b$ for every upper bound $b$ of $S$.

Claim 3. Let $t\in T$. Since $t\le t$, the number $\min\{\omega(t),M\}$ belongs to $\Omega(t)$, hence $\min\{\omega(t),M\}\le\sup\Omega(t)=\bar{\omega}(t)$. If $a\le\omega(t)$ and $a\le M$, then $a\le\min\{\omega(t),M\}$ by claim 3 of Elementary Properties of the Minimum of Two Elements, and so $a\le\bar{\omega}(t)$ by transitivity.

Claim 2. Let $s,s'\in T$ with $s\le s'$. If $t\in T$ satisfies $t\le s$, then $t\le s'$ by transitivity of $\le$, so $\Omega(s)\subseteq\Omega(s')$. Every element of $\Omega(s)$ is therefore an element of $\Omega(s')$ and is at most $\sup\Omega(s')$; thus $\sup\Omega(s')$ is an upper bound of $\Omega(s)$, and $\bar{\omega}(s)=\sup\Omega(s)\le\sup\Omega(s')=\bar{\omega}(s')$.

Claim 1. Let $s\in T$. Since $0\le\omega(0)$ by clause 1 of Modulus of Continuity and $0\le M$, claim 3 of Elementary Properties of the Minimum of Two Elements gives $0\le\min\{\omega(0),M\}$, and this number lies in $\Omega(s)$ because $0\le s$; hence $0\le\bar{\omega}(s)$. As $M$ is an upper bound of $\Omega(s)$, $\bar{\omega}(s)\le M$. Thus $\bar{\omega}$ satisfies clause 1 of Modulus of Continuity. For clause 2, let $\varepsilon\in\mathbb{R}$ be positive. By clause 2 of Modulus of Continuity for $\omega$ there is a positive $\eta\in\mathbb{R}$ such that every $t\in T$ with $t\le\eta$ satisfies $\omega(t)\le\varepsilon$. Let $s\in T$ satisfy $s\le\eta$, and let $c\in\Omega(s)$, say $c=\min\{\omega(t),M\}$ with $t\in T$ and $t\le s$. Then $t\le\eta$ by transitivity, so $\omega(t)\le\varepsilon$, and $c\le\omega(t)$ by claim 1 of Elementary Properties of the Minimum of Two Elements; hence $c\le\varepsilon$. Thus $\varepsilon$ is an upper bound of $\Omega(s)$, and $\bar{\omega}(s)\le\varepsilon$. This is clause 2 for $\bar{\omega}$, with the same $\eta$.