I am reading the book: Fully nonlinear elliptic equations of Caffarelli and Cabre. In page 8 (Prop 1.2) they prove that if function $u$ in a convex domain locally has at least one paraboloid touching from above and one touching from below then $u$ is $C^{1,1}$.
The proof basically is by proving $u$ is differentiable, and then it has weak second derivative which is bounded (I can understand this part). Now, let assume that $u$ is differentiable and has weak second derivative which is bounded, the remaining part is the following:
Since $u_i:=\partial_i u \in W^{1,\infty}(B)$ and $B$ is convex, we have that $u_i$ is continuous and:
$$u_i(x)-u_i(y)=\int_{0}^{1} \frac{d}{dt}u_i(tx+(1-t)y)\,dt=\sum\limits_{j}\int_{0}^{1} \partial_{ij}u(tx+(1-t)y)\,dt(x_j-y_j)$$
for any $x,y \in \bar{B}$.
Then using the boundedness of weak second derivative to have the Lipschitz continuity of first derivative.
My question is just they start from the assumption that $u_i \in W^{1,\infty}(B)$ and I do not know why we have this? My problem is I do not know why $u_i \in L^{\infty}(B)$. Could anyone help me please. Thank you very much!