The Sup-Convolution Converges Pointwise to an Upper Semicontinuous Function

Let $M$, $\mathbb{R}$, $\mathbb{R}^{M}$ and $\lVert\,\cdot\,\rVert$ be as in Sup-Convolution of a Function on $\mathbb{R}^M$, and let $d$ be the Euclidean distance, a metric on $\mathbb{R}^{M}$.

Let $v:\mathbb{R}^{M}\to\mathbb{R}$ be upper semicontinuous on $\mathbb{R}^{M}$ with respect to $d$, let $C\in\mathbb{R}$ be an upper bound for the set of values of $v$, and for $\lambda\in\mathbb{R}$ with $0<\lambda$ let $v^{\lambda}$ be the sup-convolution of $v$ with parameter $\lambda$.

Let $\xi\in\mathbb{R}^{M}$. Then for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\lambda_{0}\in\mathbb{R}$ with $0<\lambda_{0}$ such that every $\lambda\in\mathbb{R}$ with $\lambda_{0}\le\lambda$ satisfies

$$v(\xi)\le v^{\lambda}(\xi)<v(\xi)+\varepsilon .$$