This part is from page 107 in Michael E. Taylor's book Partial Differential Equations III.

In this part, we want a proof for the existence of smooth solution of the PDE

$\Delta u=f(x, u)$ on $U$ with boudary condition $\left.u\right|_{\partial U}=g$ where $g$ is smooth

under the assumption that $\frac{\partial f}{\partial u} \geq 0$.

After making the temporary assumption that for $|u| \geq K, \partial_{u} f(x, u)=0$ we have that

there exists soomth solution of the PDE

$\Delta u=f(x, u)$ on $U$ with boudary condition $\left.u\right|_{\partial U}=g$ when $g$ is smooth.

we construct a sequence $f_{j}(x, u)=f(x, u)$, for $|u| \leq j，f_{j}(x, u)=f(x, u)$ and when $|u| \geq K_{j}, \partial_{u} f_{j}(x, u)=0$

so we have solutions $u_{j} \in C^{\infty}(\bar{U})$ to

$\Delta u_{j}=f_{j}\left(x, u_{j}\right),\left.\quad u_{j}\right|_{\partial U}=g$

And we have $\sup _{U}\left|u_{j}\right| \leq \sup _{U} 2|\Phi|$

where $f(x, 0)=\varphi(x) \in C^{\infty}(\bar{U})$, take $g \in C^{\infty}(\partial U)$, and let $\Phi \in C^{\infty}(\bar{U})$ be the solution to

$\Delta \Phi=\varphi$ on $U, \quad \Phi=g$ on $\partial U$

Then we can see that the sequence $\left(u_{j}\right)$ stabilizes for large $j$, then we finish the proof.

How does this proof finish the proof, by Arzela-Ascoli? But this does not meet the conditions of Arzela-Ascoli theorem, actually I don't even know how to use Arzela-Ascoli here.