Let $n \in \mathbb{N}$, let $Q \subset \mathbb{R} \times \mathbb{R}^n$ be open with respect to the Euclidean topology on $\mathbb{R}^{1+n}$, and for each $k \in \mathbb{N}$ let $F_k: Q \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be continuous. Let $F: Q \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$ be continuous. Assume that $F_k \to F$ locally uniformly on $Q \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^n$, meaning that for every compact set $C \subset Q \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^n$, $\sup_{z \in C}|F_k(z)-F(z)| \to 0$ as $k \to \infty$.

Let $u_k: Q \to \mathbb{R}$ be continuous functions and let $u: Q \to \mathbb{R}$ be continuous. Assume that $u_k \to u$ locally uniformly on $Q$, meaning that for every compact set $K \subset Q$, $\sup_{(t,x) \in K}|u_k(t,x)-u(t,x)| \to 0$ as $k \to \infty$.

If, for every $k \in \mathbb{N}$, $u_k$ is a viscosity subsolution of $F_k(t,x,u,\partial_t u,\nabla u)=0$ on $Q$, meaning that for every $(t_0,x_0) \in Q$ and every [test function](/page/Test%20Function) $\phi: Q \to \mathbb{R}$ with $\phi \in C^1(Q)$ such that $u_k-\phi$ has a local maximum at $(t_0,x_0)$, one has $F_k(t_0,x_0,u_k(t_0,x_0),\partial_t\phi(t_0,x_0),\nabla\phi(t_0,x_0)) \le 0$, then $u$ is a viscosity subsolution of $F(t,x,u,\partial_t u,\nabla u)=0$ on $Q$ with the same convention.

Likewise, if, for every $k \in \mathbb{N}$, $u_k$ is a viscosity supersolution of $F_k(t,x,u,\partial_t u,\nabla u)=0$ on $Q$, meaning that for every $(t_0,x_0) \in Q$ and every test function $\phi: Q \to \mathbb{R}$ with $\phi \in C^1(Q)$ such that $u_k-\phi$ has a local minimum at $(t_0,x_0)$, one has $F_k(t_0,x_0,u_k(t_0,x_0),\partial_t\phi(t_0,x_0),\nabla\phi(t_0,x_0)) \ge 0$, then $u$ is a viscosity supersolution of $F(t,x,u,\partial_t u,\nabla u)=0$ on $Q$ with the same convention.