Proof of The Maximum of Two Viscosity Subsolutions is a Viscosity Subsolution

Conventions. Elementary order facts are those of Elementary Order Arithmetic in an Ordered Field, whose claim 1 is the compatibility of the strict order with addition, the non-strict law being an axiom of the ordered field $\mathbb{R}$. A strict inequality $a<b$ implies $a\le b$ by the definition of the strict order.

Proof. Put $\mathcal{F}=\{u,v\}$, a nonempty set of viscosity subsolutions of $F$ on $U$.

The family is locally uniformly bounded above. Let $x\in U$. Being viscosity subsolutions, $u$ and $v$ are upper semicontinuous on $U$, so, applying upper semicontinuity at $x$ with the positive real $1$, which is positive by claim 6 of Elementary Order Arithmetic in an Ordered Field, there are positive $\delta_{u},\delta_{v}\in\mathbb{R}$ such that every $y\in U$ with $d_{E}(x,y)<\delta_{u}$ satisfies $u(y)<u(x)+1$ and every $y\in U$ with $d_{E}(x,y)<\delta_{v}$ satisfies $v(y)<v(x)+1$. By claim 9 of Elementary Order Arithmetic in an Ordered Field there is a positive $\delta$ with $\delta\le\delta_{u}$ and $\delta\le\delta_{v}$, and by claim 8 of that lemma the real number $r=\tfrac{\delta}{2}$ is positive and satisfies $r<\delta$. Put $c=\max\{u(x),v(x)\}+1$. Let $y\in U$ satisfy $d_{E}(y,x)\le r$. Then $d_{E}(x,y)=d_{E}(y,x)\le r<\delta$ by the symmetry axiom of a metric, so $d_{E}(x,y)<\delta_{u}$ and $d_{E}(x,y)<\delta_{v}$ by the mixed transitivity of claim 2 of Elementary Order Arithmetic in an Ordered Field; hence $u(y)<u(x)+1\le c$ and $v(y)<v(x)+1\le c$ by claim 1 of Elementary Properties of the Maximum of Two Elements and the compatibility of $\le$ with addition. Thus every member of $\mathcal{F}$ is at most $c$ at every such $y$.

The supremum is the maximum. Let $w:U\to\mathbb{R}$ be the pointwise supremum of $\mathcal{F}$ formed in The Upper Semicontinuous Envelope of a Supremum of Viscosity Subsolutions is a Viscosity Subsolution. For $x\in U$ the set of values is $\{u(x),v(x)\}$; by claim 1 of Elementary Properties of the Maximum of Two Elements the number $\max\{u(x),v(x)\}$ is an upper bound of it, and by claim 3 of that lemma every upper bound $c'$ of it satisfies $\max\{u(x),v(x)\}\le c'$. Hence $\max\{u(x),v(x)\}$ is the least upper bound, that is $w(x)=(u\vee v)(x)$.

Conclusion. By claim 2 of The Maximum of Two Upper Semicontinuous Functions the function $u\vee v$ is upper semicontinuous on $U$, so $w$ is, and claim 4 of Properties of the Upper Semicontinuous Envelope gives $w^{*}(x)=w(x)$ for every $x\in U$; that is, $w^{*}=u\vee v$. By The Upper Semicontinuous Envelope of a Supremum of Viscosity Subsolutions is a Viscosity Subsolution the function $w^{*}$ is a viscosity subsolution of $F$ on $U$, and therefore so is $u\vee v$. $\blacksquare$