A proof for the existence of smooth solution of PDE in form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III

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.

Since you have the uniform bound $$\sup_U |u_j| \leq \sup_U 2|\Phi|$$ For simplicity I will assume your $$K_j = j$$. Take $$J = \sup_U 2|\Phi|$$, then for every $$j,k \geq J$$ you have that $$f_j(x,u_j) = f_k(x,u_j) = f(x,u_j).$$ This tells you that for every $$j,k \geq J$$ you have $$\Delta u_j = f_k(x,u_j).$$ By the uniqueness of the solution to the problem with the temporary truncation, you find therefore that $$u_k = u_j$$ for every $$j,k\geq J$$.
(This is what the author means by "stabilizes for large $$j$$"; that after some $$J$$ the sequence $$\{u_j\}$$ becomes the constant sequence.)