[proofplan] For each requested precision $2^{-k}$, we use the finite local data supplied at level $k$. The modulus exponent $\omega(k)$ is chosen to dominate both the Lebesgue exponent $\ell(k)$ and all local exponents $\eta_{k,i}$. Then any two points within distance $2^{-\omega(k)}$ lie in a common interval of the finite cover, and on that interval the local estimate applies. The final inequality follows from the harmless stronger bound $2^{-(k+1)} \le 2^{-k}$. [/proofplan]

[step:Define the candidate modulus by taking the maximum of the finite exponents] Fix $k \in \mathbb{N}$. The hypotheses provide a finite cover $I_{k,1},\dots,I_{k,N_k}$, local exponents $\eta_{k,1},\dots,\eta_{k,N_k} \in \mathbb{N}$, and a Lebesgue exponent $\ell(k) \in \mathbb{N}$. Define \begin{align*} \omega(k) := \max\{\ell(k), \eta_{k,1}, \dots, \eta_{k,N_k}\}. \end{align*} Since the displayed set is finite and nonempty, $\omega(k) \in \mathbb{N}$ is well-defined. Doing this for each $k \in \mathbb{N}$ defines a map \begin{align*} \omega: \mathbb{N} \to \mathbb{N}. \end{align*} By construction, for every $i \in \{1,\dots,N_k\}$, \begin{align*} \omega(k) \ge \ell(k) \end{align*} and \begin{align*} \omega(k) \ge \eta_{k,i}. \end{align*} Because the function $m \mapsto 2^{-m}$ is decreasing on $\mathbb{N}$, these inequalities imply \begin{align*} 2^{-\omega(k)} \le 2^{-\ell(k)} \end{align*} and \begin{align*} 2^{-\omega(k)} \le 2^{-\eta_{k,i}}. \end{align*} [/step]

[step:Use the Lebesgue exponent to place close points in one interval]Let $x,y \in [a,b]$ satisfy \begin{align*} |x-y| \le 2^{-\omega(k)}. \end{align*} Since $2^{-\omega(k)} \le 2^{-\ell(k)}$, we have \begin{align*} |x-y| \le 2^{-\ell(k)}. \end{align*} By the Lebesgue exponent hypothesis for the cover $I_{k,1},\dots,I_{k,N_k}$, there exists an index $i \in \{1,\dots,N_k\}$ such that \begin{align*} x,y \in I_{k,i}. \end{align*} Since also $x,y \in [a,b]$, this gives \begin{align*} x,y \in [a,b] \cap I_{k,i}. \end{align*}[/step]

[guided]We now explain why the maximum chosen in the previous step is the right exponent. We start with arbitrary points $x,y \in [a,b]$ satisfying \begin{align*} |x-y| \le 2^{-\omega(k)}. \end{align*} The definition of $\omega(k)$ included the Lebesgue exponent $\ell(k)$ among the numbers being maximized, so $\omega(k) \ge \ell(k)$. Since larger exponents give smaller dyadic radii, this means \begin{align*} 2^{-\omega(k)} \le 2^{-\ell(k)}. \end{align*} Combining the two inequalities gives \begin{align*} |x-y| \le 2^{-\ell(k)}. \end{align*} The role of $\ell(k)$ is precisely to convert closeness of two points into membership in one common member of the cover. Therefore, by the Lebesgue exponent hypothesis, there is an index $i \in \{1,\dots,N_k\}$ such that both points belong to the same rational interval: \begin{align*} x,y \in I_{k,i}. \end{align*} Because the points were originally chosen from $[a,b]$, we may record the stronger membership condition needed for the local estimate: \begin{align*} x,y \in [a,b] \cap I_{k,i}. \end{align*} This is the only place where the Lebesgue number condition is used: it turns a global small-distance assumption into the local situation where one of the interval estimates applies.[/guided]

[step:Apply the local estimate on the selected interval] For the index $i$ obtained above, the definition of $\omega(k)$ gives $\omega(k) \ge \eta_{k,i}$, hence \begin{align*} 2^{-\omega(k)} \le 2^{-\eta_{k,i}}. \end{align*} Together with $|x-y| \le 2^{-\omega(k)}$, this yields \begin{align*} |x-y| \le 2^{-\eta_{k,i}}. \end{align*} Since $x,y \in [a,b] \cap I_{k,i}$, the local estimate for the interval $I_{k,i}$ gives \begin{align*} |f(x)-f(y)| \le 2^{-(k+1)}. \end{align*} Finally, \begin{align*} 2^{-(k+1)} \le 2^{-k}, \end{align*} so \begin{align*} |f(x)-f(y)| \le 2^{-k}. \end{align*} [/step]

[step:Conclude that the constructed function is a modulus of uniform continuity] We have shown that for every $k \in \mathbb{N}$ and all $x,y \in [a,b]$, \begin{align*} |x-y| \le 2^{-\omega(k)} \implies |f(x)-f(y)| \le 2^{-k}. \end{align*} Thus the map $\omega: \mathbb{N} \to \mathbb{N}$ is a modulus of [uniform continuity](/page/Uniform%20Continuity) for $f$ on $[a,b]$. The value $\omega(k)$ is computed from the finite data supplied at precision $k$, namely $\ell(k)$ and the finite list $\eta_{k,1},\dots,\eta_{k,N_k}$, so the construction is uniform in the stated constructive sense. [/step]