I actually posted this on math.stackexchange but it wasn't getting responses even after a bounty. I thought maybe it is too specialized so I'll post it here. I'm currently reading the user's guide to viscosity solutions. In Lemma 4.2 we define $w(x) = \sup\{u(x):u\in\mathcal{F}\}$, where $\mathcal{F}$ is a family of subsolutions to a certain equation. Next we consider the upper-semicontinuous envelope $w^*(x)$ and the lemma claims that $w^*$ is a subsolution as well. So far so good.

Later in section 6, we encounter a similar construction as before. Given a sequence $u_n(x)$ of subsolutions to an equation, we define the 'limit' $\bar{U}(z)=\limsup_{j\to\infty}\{u_n(x):n\geq j, |z-x|\leq\frac{1}{j}\}$, that is, we take the limsup and * operation simultaneously, instead of limsupping followed by * as before. Lemma 6.1 then claims that $\bar{U}(z)$ is a subsolution as well.

My question is, what is the difference between these two constructions; are there examples of sequences of solutions of functions whose 'limits' in the lemma 4.2 sense and the lemma 6.1 sense are different?