The Maximum of Two Viscosity Subsolutions is a Viscosity Subsolution

Throughout we work in the setting of Second-Order Equations on Euclidean Open Sets, whose notation, including that of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation and Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation on which it rests, is in force in a dimension $n$, a natural number with $1\le n$.

Let $U\subseteq\mathbb{R}^{n}$ be open and nonempty, let $F$ be a second-order equation operator on $U$ that is continuous, and let $u,v:U\to\mathbb{R}$ both be viscosity subsolutions of $F$ on $U$. Let $u\vee v:U\to\mathbb{R}$ be the function whose value at $x$ is the maximum $\max\{u(x),v(x)\}$, as in The Maximum of Two Upper Semicontinuous Functions.

Then $u\vee v$ is a viscosity subsolution of $F$ on $U$.