Classical Sub- and Supersolutions of a Degenerate Elliptic Operator are Viscosity Sub- and Supersolutions

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $U\subseteq\mathbb{R}^{n}$ be open, let $F$ be a second-order equation operator on $U$ that is degenerate elliptic, and let $u:U\to\mathbb{R}$ be of class $C^{2}$ on $U$.

Then the following hold.

1. (Subsolutions) If $u$ is a classical subsolution of $F$ on $U$, then $u$ is a viscosity subsolution of $F$ on $U$.

2. (Supersolutions) If $u$ is a classical supersolution of $F$ on $U$, then $u$ is a viscosity supersolution of $F$ on $U$.