[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*}
[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]
[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]
