The upper envelope of a locally bounded supremum of subsolutions is a subsolution

Strict-contact case. Let $\varphi \in C^1(U)$ touch $w^{\ast }$ from above at $z_0=(x_0,t_0)$ with a strict local maximum of $w^{\ast }-\varphi$, and suppose $\theta :={\varphi }_t(z_0)+H(z_0,D\varphi (z_0))>0$. Choose $r>0$ so that $\bar{B}(z_0,r)\subseteq U$, the contact is strict on this ball, and ${\varphi }_t(z)+2(t-t_0)+H(z,D\varphi (z)+2(x-x_0))>\theta \mathrm{/}2$ throughout it, by continuity. On the compact annulus $A=\{r\mathrm{/}2\le \mathrm{\mid }z-z_0\mathrm{\mid }\le r\}$, [F1, F3] give the positive gap $g=(w^{\ast }-\varphi )(z_0)-{\max }_A(w^{\ast }-\varphi )$. Choose $0<\eta <g\mathrm{/}4$ and $0<\delta <r\mathrm{/}2$ such that $\mathrm{\mid }\varphi (y)-\varphi (z_0)\mathrm{\mid }<\eta$ and $\mathrm{\mid }y-z_0{\mathrm{\mid }}^2<\eta$ for $\mathrm{\mid }y-z_0\mathrm{\mid }<\delta$. The two supremum definitions in [F1] supply one pair $(w^{\mathrm{\prime }},y)$ with $w^{\mathrm{\prime }}\in \mathcal{F}$, $\mathrm{\mid }y-z_0\mathrm{\mid }<\delta$, and $w^{\mathrm{\prime }}(y)>w^{\ast }(z_0)-\eta$ (use a closed radius smaller than $\delta$). By [F3], $w^{\mathrm{\prime }}-\varphi -\mathrm{\mid }z-z_0{\mathrm{\mid }}^2$ attains a maximum on $\bar{B}(z_0,r)$, of value greater than $(w^{\ast }-\varphi )(z_0)-3\eta$. Its value on $A$ is at most $(w^{\ast }-\varphi )(z_0)-g$, since $w^{\mathrm{\prime }}\le w\le w^{\ast }$. Thus any maximiser $z_{\ast }=(x_{\ast },t_{\ast })$ lies in $B(z_0,r\mathrm{/}2)$ and is an interior upper contact for $\psi =\varphi +\mathrm{\mid }z-z_0{\mathrm{\mid }}^2$. Its derivatives are ${\psi }_t(z_{\ast })={\varphi }_t(z_{\ast })+2(t_{\ast }-t_0)$ and $D\psi (z_{\ast })=D\varphi (z_{\ast })+2(x_{\ast }-x_0)$. Their residual is greater than $\theta \mathrm{/}2$, contradicting the subsolution inequality [F2]. Hence the desired residual at $z_0$ is nonpositive.

General contacts and conclusion. If $w^{\ast }-\varphi$ merely has a local maximum at $z_0$, fix $r>0$ with $\bar{B}(z_0,r)\subseteq U$ on which the maximum inequality $w^{\ast }-\varphi \le w^{\ast }(z_0)-\varphi (z_0)$ holds and strictify by [F4]: the test ${\varphi }_{\epsilon }=\varphi +\epsilon \mathrm{\mid }z-z_0{\mathrm{\mid }}^4$ has the same value and first jet at $z_0$ and makes $w^{\ast }-{\varphi }_{\epsilon }$ strictly maximised at $z_0$ over $\bar{B}(z_0,r)$. Step 1.1 applied to ${\varphi }_{\epsilon }$ gives ${\varphi }_t(z_0)+H(z_0,D\varphi (z_0))=({\varphi }_{\epsilon }{)}_t(z_0)+H(z_0,D{\varphi }_{\epsilon }(z_0))\le 0$. Hence $w^{\ast }$ is a viscosity subsolution of the equation in $U$; the selection of the single witness $(w^{\mathrm{\prime }},y)$ and of the compact maximiser $z_{\ast }$ involves no choice principle, and the whole argument is pointwise.