Proof of Closed Superlevel Sets of a Sum on a Product Space and of a Doubled Function

Each result cited is universally quantified over the data in its own statement. Let $K_{1},K_{2}\in\mathbb{R}$ satisfy $u_{1}(x)\le K_{1}$ for every $x\in S_{1}$ and $u_{2}(y)\le K_{2}$ for every $y\in S_{2}$, and write $u(x,y)=u_{1}(x)+u_{2}(y)$ for $(x,y)\in S_{1}\times S_{2}$.

Claim 1. We verify the condition of Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits §sequential in the metric space $E_{1}\times E_{2}$. Let $\bigl((x_{m},y_{m})\bigr)_{m\in\mathbb{N}}$ be a sequence in $S_{1}\times S_{2}$ converging in $E_{1}\times E_{2}$ to a point $(x,y)$, and let $t\in\mathbb{R}$ satisfy $t\le u(x_{m},y_{m})$ for every $m\in\mathbb{N}$. By Properties of the Product of Two Real Inner Product Spaces §componentwise the sequence $(x_{m})_{m\in\mathbb{N}}$ converges to $x$ in $E_{1}$ and $(y_{m})_{m\in\mathbb{N}}$ converges to $y$ in $E_{2}$.

Since $u_{2}(y_{m})\le K_{2}$, we get $t-K_{2}\le u_{1}(x_{m})$ for every $m$, so Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits §sequential, applied to $u_{1}$, gives $x\in S_{1}$; symmetrically $y\in S_{2}$, so $(x,y)\in S_{1}\times S_{2}$.

By Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits §usc the function $u_{1}$ is upper semicontinuous on $S_{1}$ and $u_{2}$ is upper semicontinuous on $S_{2}$. Let $\epsilon\in\mathbb{R}$ be positive. By upper semicontinuity at $x$ relative to $S_{1}$ there is a positive $\rho_{1}\in\mathbb{R}$ such that every $x'\in S_{1}$ with $d_{E_{1}}(x',x)<\rho_{1}$ satisfies $u_{1}(x')\le u_{1}(x)+\epsilon$, and similarly there is a positive $\rho_{2}$ for $u_{2}$ at $y$. By the convergence of the two coordinate sequences there is $m\in\mathbb{N}$ with $d_{E_{1}}(x_{m},x)<\rho_{1}$ and $d_{E_{2}}(y_{m},y)<\rho_{2}$, and for that $m$

As $\epsilon$ ranges over all positive reals so does $2\epsilon$, so Comparison of Real Numbers with Arbitrary Positive Slack gives $t\le u_{1}(x)+u_{2}(y)=u(x,y)$. By Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits §sequential again, $u$ has closed superlevel sets in $E_{1}\times E_{2}$.

Claim 2. Let $q:H\times H\to\mathbb{R}$ be given by $q(x,y)=\tfrac{\alpha}{2}|x-y|_{H}^{2}$. By The Doubling Form on the Product of a Real Hilbert Space with Itself §c2 the function $q$ belongs to $C^{2}(H\times H)$; in particular it is differentiable at every point of $H\times H$, hence continuous on $H\times H$ by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §continuous. Since $\Phi(x,y)=u(x,y)-q(x,y)$ for $(x,y)\in S_{1}\times S_{2}$, claim 1 together with Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits §perturbation shows that $\Phi$ has closed superlevel sets in $H\times H$.

Assume now the stated bounds and $0\le\alpha$. For $(x,y)\in S_{1}\times S_{2}$ we have $0\le\tfrac{\alpha}{2}|x-y|_{H}^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field, hence

the last equality by the formula for the norm of the product in Properties of the Product of Two Real Inner Product Spaces §norm.