Proof of Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy on Bounded Noncommutative Laws

Each result cited is applied with the data of its own statement. Throughout, $R>0$ is the wall radius of The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §parameters, $\mathbb{N}=\{1,2,3,\dots\}$ as in Natural Numbers, and $\iota=\iota_{\mathbb{R}}:\mathbb{N}\to\mathbb{R}$ is the canonical map of The Canonical Map from the Natural Numbers to a Field. For $m\in\mathbb{N}$ we have $0<\iota(m)$ by claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so the inverse $\iota(m)^{-1}$ exists and is positive by claim 7 of Elementary Order Arithmetic in an Ordered Field; we write $1/m:=\iota(m)^{-1}$, a positive real number. Moreover, for $m,n\in\mathbb{N}$ with $m\le n$ we have $\iota(m)\le\iota(n)$: by Order on the Natural Numbers, $m\le n$ means $m<n$ or $m=n$; in the first case $\iota(m)<\iota(n)$ by claim 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and in the second $\iota(m)=\iota(n)$. For reals $\delta\ge0$ and $\alpha>0$ let $V(\delta,\alpha)=\{\Psi_{\delta,\alpha}(\mu,\nu):\ (\mu,\nu)\in\mathcal{D}\times\mathcal{D}\}$, so that $S(\delta,\alpha)=\sup V(\delta,\alpha)$.

Step 1 (Preliminaries). (a) By The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation §free-energy, $\mathcal{D}$ and $\mathcal{E}$ are the domain and the energy of The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score for the free entropy penalty $(\mathcal{D}_{0},\mathcal{E}_{0})$ and the radius $R$. Hence $\mathcal{D}\subseteq\Sigma_{d,R}$ by The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §bounds, and $\Sigma_{d,R}\subseteq\Sigma_{d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law.

(b) For $\mu,\nu\in\mathcal{D}$ the number $W_{2}(\mu,\nu)$ is a nonnegative square root by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance, so $W_{2}(\mu,\nu)\ge0$, and then $0=W_{2}(\mu,\nu)\cdot0\le W_{2}(\mu,\nu)\cdot W_{2}(\mu,\nu)=W_{2}(\mu,\nu)^{2}$ by claim 1 of Zero Products and Elementary Identities in a Field and claim 5 of Elementary Arithmetic in an Ordered Field. In the same way, whenever $0\le a$ and $0\le x$ are reals, $0=a\cdot0\le ax$ by these two claims; we use this below for products of nonnegative factors.

(c) If a real sequence $(x_{k})_{k\in\mathbb{N}}$ converges to $L$ and $(k_{j})_{j\in\mathbb{N}}$ is strictly increasing in $\mathbb{N}$, then $(x_{k_{j}})_{j\in\mathbb{N}}$ converges to $L$: given $\varepsilon>0$, take $N$ as in Limit of a Sequence of Real Numbers for $(x_{k})$; every $j\ge N$ has $k_{j}\ge j\ge N$ by Strictly Increasing Sequences of Natural Numbers Dominate Their Index, hence $|x_{k_{j}}-L|<\varepsilon$.

(d) Let a real sequence $(a_{l})_{l\in\mathbb{N}}$ converge to $a$, let $c'$ be real and $l_{0}\in\mathbb{N}$. If $a_{l}\le c'$ for every $l\ge l_{0}$, then $a\le c'$; if $a_{l}\ge c'$ for every $l\ge l_{0}$, then $a\ge c'$. Indeed, in the first case suppose $c'<a$ (the order is total) and put $\varepsilon=a-c'>0$; with $N$ as in Limit of a Sequence of Real Numbers and $l=N+l_{0}$ we get $|a_{l}-a|<\varepsilon$, so $a_{l}>a-\varepsilon=c'$ by claim 9 of Properties of the Absolute Value in an Ordered Field, a contradiction. In the second case suppose $a<c'$, put $\varepsilon=c'-a$, and obtain in the same way $a_{l}<a+\varepsilon=c'$, a contradiction.

(e) For $\delta\ge0$ and $\lambda\in\mathcal{D}$ we have $\delta e\le\delta\,\mathcal{E}(\lambda)$ by the choice of $e$ and claim 5 of Elementary Arithmetic in an Ordered Field; and for $\lambda,\lambda'\in\Sigma^{2}_{d}$ we have $u(\lambda)\le|u(\lambda)|\le b$ by claim 3 of Properties of the Absolute Value in an Ordered Field, and $-v(\lambda')\le|-v(\lambda')|=|v(\lambda')|\le b$ by claim 3 of Properties of the Absolute Value in an Ordered Field applied to $-v(\lambda')$ and claim 2 of Properties of the Absolute Value in an Ordered Field. We also record: $0<2$ by claim 8 of Elementary Order Arithmetic in an Ordered Field, so for every real $\alpha>0$ the number $\tfrac{\alpha}{2}=\alpha\cdot2^{-1}$ is positive by that claim 8, and in particular $0\le\tfrac{\alpha}{2}$.

Step 2 (Clause bounds). Let $\delta\ge0$, $\alpha>0$, $\mu,\nu\in\mathcal{D}$. By Step 1(e), $u(\kappa_{d}(\mu))\le b$, $-v(\kappa_{d}(\nu))\le b$, $-\delta\mathcal{E}(\mu)\le-\delta e$ and $-\delta\mathcal{E}(\nu)\le-\delta e$; by Step 1(b), the positivity of $\tfrac{\alpha}{2}$ recorded in Step 1(e), and the product rule for nonnegative factors of Step 1(b), $\tfrac{\alpha}{2}W_{2}(\mu,\nu)^{2}\ge0$. Adding, $\Psi_{\delta,\alpha}(\mu,\nu)\le2b-2\delta e$. Since $\mathcal{D}\ne\varnothing$, $V(\delta,\alpha)$ is nonempty, and $2b-2\delta e$ is an upper bound for it in the sense of Upper Bound and Least Upper Bound, so $V(\delta,\alpha)$ is bounded above; hence its least upper bound $S(\delta,\alpha)$ exists as a real number, since $\mathbb{R}$ is Dedekind complete by The Real Numbers and Standard Notation. Moreover, for $\mu,\nu\in\mathcal{D}$,

$$\Psi_{\delta,\alpha}(\mu,\nu)=\Psi_{0,\alpha}(\mu,\nu)-\delta\mathcal{E}(\mu)-\delta\mathcal{E}(\nu)\le S(0,\alpha)-2\delta e,$$

because $S(0,\alpha)$ is an upper bound of $V(0,\alpha)$ and by Step 1(e). Thus $S(0,\alpha)-2\delta e$ is an upper bound of $V(\delta,\alpha)$, and the least upper bound satisfies $S(\delta,\alpha)\le S(0,\alpha)-2\delta e$.

Step 3 (Clause diagonal). Let $\alpha>0$ and $\lambda\in\mathcal{D}$. By Step 1(a), $\lambda\in\Sigma_{d,R}$, so $W_{2}(\lambda,\lambda)=0$ by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §separation. Hence $\Psi_{0,\alpha}(\lambda,\lambda)=u(\kappa_{d}(\lambda))-v(\kappa_{d}(\lambda))$, and this value is at most $S(0,\alpha)$ since it lies in $V(0,\alpha)$.

Step 4 (Clause monotone). Let $0<\alpha'\le\alpha$ and $\mu,\nu\in\mathcal{D}$. Since $\alpha'\le\alpha$, claim 3 of Elementary Arithmetic in an Ordered Field gives $0\le\alpha-\alpha'$; and $0<2^{-1}$ by claims 8 and 7 of Elementary Order Arithmetic in an Ordered Field. Hence $\tfrac{\alpha-\alpha'}{2}=2^{-1}(\alpha-\alpha')\ge0$ by the product rule for nonnegative factors of Step 1(b). As $W_{2}(\mu,\nu)^{2}\ge0$ (Step 1(b)), the same rule gives $\tfrac{\alpha-\alpha'}{2}W_{2}(\mu,\nu)^{2}\ge0$, so

$$\Psi_{0,\alpha}(\mu,\nu)=\Psi_{0,\alpha'}(\mu,\nu)-\tfrac{\alpha-\alpha'}{2}W_{2}(\mu,\nu)^{2}\le\Psi_{0,\alpha'}(\mu,\nu)\le S(0,\alpha').$$

So $S(0,\alpha')$ is an upper bound of $V(0,\alpha)$, whence $S(0,\alpha)\le S(0,\alpha')$.

Step 5 (Clause maximiser: an approximating sequence). Fix reals $\delta>0$ and $\alpha>0$, and write $\Psi=\Psi_{\delta,\alpha}$ and $S=S(\delta,\alpha)$, a real number by Step 2. For each $m\in\mathbb{N}$, claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ with $\varepsilon=1/m$ gives $(\mu_{m},\nu_{m})\in\mathcal{D}\times\mathcal{D}$ with

$$S-1/m<\Psi(\mu_{m},\nu_{m}).\tag{5.1}$$

Write $w_{m}=W_{2}(\mu_{m},\nu_{m})\ge0$ (Step 1(b)). We have $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field and $1\le\iota(m)$ by claim 2 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, so Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal with $a=1$ and $b=\iota(m)$ gives $1/m=\iota(m)^{-1}\le1^{-1}=1$. Hence $-1\le-1/m$ by claim 4 of Elementary Order Arithmetic in an Ordered Field, so $S-1\le S-1/m$ by the compatibility of the order with addition in Ordered Field, and $S-1<\Psi(\mu_{m},\nu_{m})$ by (5.1) and claim 2 of Elementary Order Arithmetic in an Ordered Field. Solving the definition of $\Psi(\mu_{m},\nu_{m})$ for one term and using the bounds of Step 1(e) together with $\tfrac{\alpha}{2}w_{m}^{2}\ge0$ (as in Step 2), we get

$$\tfrac{\alpha}{2}w_{m}^{2}=u(\kappa_{d}(\mu_{m}))-v(\kappa_{d}(\nu_{m}))-\delta\mathcal{E}(\mu_{m})-\delta\mathcal{E}(\nu_{m})-\Psi(\mu_{m},\nu_{m})\le2b-2\delta e-S+1,$$ $$\delta\mathcal{E}(\mu_{m})=u(\kappa_{d}(\mu_{m}))-v(\kappa_{d}(\nu_{m}))-\tfrac{\alpha}{2}w_{m}^{2}-\delta\mathcal{E}(\nu_{m})-\Psi(\mu_{m},\nu_{m})\le2b-\delta e-S+1,$$

and in the same way $\delta\mathcal{E}(\nu_{m})\le2b-\delta e-S+1$. The numbers $\tfrac{2}{\alpha}=2\alpha^{-1}$ and $\tfrac{1}{\delta}=\delta^{-1}$ are positive: $0<\alpha^{-1}$ and $0<\delta^{-1}$ by claim 7 of Elementary Order Arithmetic in an Ordered Field, $0<2$ by claim 8 there, and $0<2\alpha^{-1}$ by claim 5 there; in particular both are nonnegative. Multiplying the first inequality by $\tfrac{2}{\alpha}$ and the other two by $\tfrac{1}{\delta}$, which preserves non-strict inequalities by claim 5 of Elementary Arithmetic in an Ordered Field, and using $\tfrac{2}{\alpha}\cdot\tfrac{\alpha}{2}=(2\alpha^{-1})(\alpha2^{-1})=1$ and $\tfrac{1}{\delta}\cdot\delta=1$ on the left-hand sides, gives, for every $m\in\mathbb{N}$,

$$w_{m}^{2}\le C,\qquad \mathcal{E}(\mu_{m})\le c,\qquad \mathcal{E}(\nu_{m})\le c,\qquad C=\tfrac{2}{\alpha}(2b-2\delta e-S+1),\quad c=\tfrac{1}{\delta}(2b-\delta e-S+1).$$

Finally $|w_{m}|=w_{m}$ by Absolute Value in an Ordered Field, since $0\le w_{m}$, and $w_{m}\le1+|C|$: if $w_{m}\le1$ this holds because $0\le|C|$; if $1<w_{m}$, then $w_{m}=w_{m}\cdot1<w_{m}\cdot w_{m}\le C\le|C|<1+|C|$ by claim 10 of Elementary Order Arithmetic in an Ordered Field (with the positive factor $w_{m}$) and claim 3 of Properties of the Absolute Value in an Ordered Field.

Step 6 (Extraction). By Step 5 and The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §compact applied to $(\mu_{m})_{m\in\mathbb{N}}$ and $c$, there are a strictly increasing $(m_{i})_{i\in\mathbb{N}}$, $\hat{\mu}\in\mathcal{D}$ and a real $L_{1}$ with $\mu_{m_{i}}\to\hat{\mu}$ weak-star, $\mathcal{E}(\mu_{m_{i}})\to L_{1}$ and $\mathcal{E}(\hat{\mu})\le L_{1}$. The sequence $(\nu_{m_{i}})_{i\in\mathbb{N}}$ lies in $\mathcal{D}$ with $\mathcal{E}(\nu_{m_{i}})\le c$, so The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score §compact applied to it gives a strictly increasing $(i_{j})_{j\in\mathbb{N}}$, $\hat{\nu}\in\mathcal{D}$ and a real $L_{2}$ with $\nu_{m_{i_{j}}}\to\hat{\nu}$ weak-star, $\mathcal{E}(\nu_{m_{i_{j}}})\to L_{2}$ and $\mathcal{E}(\hat{\nu})\le L_{2}$. Put $k_{j}=m_{i_{j}}$, strictly increasing by claim 2 of A Subsequence of a Subsequence is a Subsequence.

Apply The Diagonal Subsequence Lemma for Bounded Real Arrays to the array with $a_{j,1}=u(\kappa_{d}(\mu_{k_{j}}))$, $a_{j,2}=v(\kappa_{d}(\nu_{k_{j}}))$, $a_{j,3}=w_{k_{j}}$ and $a_{j,k}=0$ for $k\ge4$, with bounds $R_{1}=R_{2}=b$, $R_{3}=1+|C|$ and $R_{k}=0$ for $k\ge4$ (valid by hypothesis and Step 5). It gives a strictly increasing $(j_{l})_{l\in\mathbb{N}}$ and reals $A$, $B$, $w$ such that, with $n_{l}=k_{j_{l}}=m_{i_{j_{l}}}$,

$$A_{l}:=u(\kappa_{d}(\mu_{n_{l}}))\to A,\qquad B_{l}:=v(\kappa_{d}(\nu_{n_{l}}))\to B,\qquad w_{n_{l}}\to w .$$

By claim 2 of A Subsequence of a Subsequence is a Subsequence, $(i_{j_{l}})_{l}$ and $(n_{l})_{l}$ are strictly increasing, and by its claim 3, $(\mu_{n_{l}})_{l}$ is the subsequence of $(\mu_{m_{i}})_{i}$ determined by $(i_{j_{l}})_{l}$, and $(\nu_{n_{l}})_{l}$ is the subsequence of $(\nu_{m_{i_{j}}})_{j}$ determined by $(j_{l})_{l}$. Hence, by Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §unique and Step 1(c),

$$\mu_{n_{l}}\to\hat{\mu}\ \text{and}\ \nu_{n_{l}}\to\hat{\nu}\ \text{weak-star},\qquad \mathcal{E}(\mu_{n_{l}})\to L_{1},\qquad \mathcal{E}(\nu_{n_{l}})\to L_{2}.$$

All of $\mu_{n_{l}}$, $\nu_{n_{l}}$, $\hat{\mu}$, $\hat{\nu}$ lie in $\Sigma_{d,R}\subseteq\Sigma_{d}$ by Step 1(a).

Step 7 (Comparison of the limits). (i) $A\le u(\kappa_{d}(\hat{\mu}))$. Fix $\varepsilon>0$ first and put $c'=u(\kappa_{d}(\hat{\mu}))+\varepsilon>u(\kappa_{d}(\hat{\mu}))$. By Weak-Star Semicontinuity on Bounded Laws of Functions on Square-Integrable Noncommutative Laws §upper, applied with the radius $R$ to $(\mu_{n_{l}})_{l}$ and $\hat{\mu}$, there is $l_{0}$ with $A_{l}<c'$ for all $l\ge l_{0}$, so $A\le u(\kappa_{d}(\hat{\mu}))+\varepsilon$ by Step 1(d). As $\varepsilon>0$ was arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack §slack-above gives the claim.

(ii) $v(\kappa_{d}(\hat{\nu}))\le B$. Fix $\varepsilon>0$ first and put $c'=v(\kappa_{d}(\hat{\nu}))-\varepsilon<v(\kappa_{d}(\hat{\nu}))$. By Weak-Star Semicontinuity on Bounded Laws of Functions on Square-Integrable Noncommutative Laws §lower, applied with the radius $R$ to $(\nu_{n_{l}})_{l}$ and $\hat{\nu}$, there is $l_{0}$ with $B_{l}>c'$ for all $l\ge l_{0}$, so $v(\kappa_{d}(\hat{\nu}))-\varepsilon\le B$ by Step 1(d), and Comparison of Real Numbers with Arbitrary Positive Slack §slack-below gives the claim.

(iii) $\hat{W}^{2}\le w^{2}$, where $\hat{W}=W_{2}(\hat{\mu},\hat{\nu})$. Fix $\varepsilon>0$ first. Since $w_{n_{l}}\to w$, there is $l_{0}$ with $|w_{n_{l}}-w|<\varepsilon$, hence $w_{n_{l}}<w+\varepsilon$ (claim 9 of Properties of the Absolute Value in an Ordered Field), for all $l\ge l_{0}$. The index sequence $(l_{0}+i)_{i\in\mathbb{N}}$ is strictly increasing, so the tails $(\mu_{n_{l_{0}+i}})_{i\in\mathbb{N}}$ and $(\nu_{n_{l_{0}+i}})_{i\in\mathbb{N}}$ are subsequences of $(\mu_{n_{l}})_{l}$ and $(\nu_{n_{l}})_{l}$ and converge weak-star to $\hat{\mu}$ and $\hat{\nu}$ by Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §unique; they lie in $\Sigma_{d,R}$, and $W_{2}(\mu_{n_{l_{0}+i}},\nu_{n_{l_{0}+i}})=w_{n_{l_{0}+i}}\le w+\varepsilon$ for every $i$. By The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §lsc, $\hat{W}\le w+\varepsilon$. As $\varepsilon$ was arbitrary, $\hat{W}\le w$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above. Since $0\le\hat{W}$ (Step 1(b)), also $0\le w$, and two applications of claim 5 of Elementary Arithmetic in an Ordered Field (with the nonnegative factors $\hat{W}$ and then $w$) give $\hat{W}\hat{W}\le\hat{W}w\le ww$.

(iv) $\mathcal{E}(\hat{\mu})\le L_{1}$ and $\mathcal{E}(\hat{\nu})\le L_{2}$ by Step 6.

Step 8 (The limit of the values). For every $l$,

$$\Psi(\mu_{n_{l}},\nu_{n_{l}})=A_{l}-B_{l}-\tfrac{\alpha}{2}w_{n_{l}}^{2}-\delta\mathcal{E}(\mu_{n_{l}})-\delta\mathcal{E}(\nu_{n_{l}}),$$

so by claims 1, 2 and 3 of Arithmetic of Limits of Real Sequences and Step 6 this sequence converges to

$$P=A-B-\tfrac{\alpha}{2}w^{2}-\delta L_{1}-\delta L_{2}.$$

We show $P\ge S$. Fix $\varepsilon>0$ first; by claim 3 of The Archimedean Property of the Real Numbers choose $N_{0}\in\mathbb{N}$ with $1/N_{0}=\iota(N_{0})^{-1}<\varepsilon$. For $l\ge N_{0}$ we have $n_{l}\ge l\ge N_{0}$ by Strictly Increasing Sequences of Natural Numbers Dominate Their Index, so $\iota(N_{0})\le\iota(n_{l})$ by the preamble, where also $0<\iota(N_{0})$; hence Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal with $a=\iota(N_{0})$ and $b=\iota(n_{l})$ gives $1/n_{l}=\iota(n_{l})^{-1}\le\iota(N_{0})^{-1}=1/N_{0}<\varepsilon$, and (5.1) with $m=n_{l}$ gives $\Psi(\mu_{n_{l}},\nu_{n_{l}})>S-1/n_{l}>S-\varepsilon$. By Step 1(d), $P\ge S-\varepsilon$; as $\varepsilon$ was arbitrary, $P\ge S$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-below.

Step 9 (Conclusion of clause maximiser). By Step 7, the sign reversal of claim 4 of Elementary Order Arithmetic in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field (with the nonnegative factors $\tfrac{\alpha}{2}$, nonnegative by Step 1(e), and $\delta>0$),

$$u(\kappa_{d}(\hat{\mu}))\ge A,\quad -v(\kappa_{d}(\hat{\nu}))\ge-B,\quad -\tfrac{\alpha}{2}\hat{W}^{2}\ge-\tfrac{\alpha}{2}w^{2},\quad -\delta\mathcal{E}(\hat{\mu})\ge-\delta L_{1},\quad -\delta\mathcal{E}(\hat{\nu})\ge-\delta L_{2}.$$

Adding, $\Psi(\hat{\mu},\hat{\nu})\ge P\ge S$ by Step 8. On the other hand $\Psi(\hat{\mu},\hat{\nu})\le S$ because $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ and $S$ is an upper bound of $V(\delta,\alpha)$. Hence $\Psi(\hat{\mu},\hat{\nu})=S$, and $(\hat{\mu},\hat{\nu})$ is a maximising pair.

Step 10 (Clause strength). Let $\delta>0$, $0<\alpha'<\alpha$, and let $(\hat{\mu},\hat{\nu})$ be a maximising pair of $\Psi_{\delta,\alpha}$. Directly from the definition,

$$\Psi_{\delta,\alpha'}(\hat{\mu},\hat{\nu})=\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})+\tfrac{\alpha-\alpha'}{2}W_{2}(\hat{\mu},\hat{\nu})^{2}=S(\delta,\alpha)+\tfrac{\alpha-\alpha'}{2}W_{2}(\hat{\mu},\hat{\nu})^{2},$$

and the left side is at most $S(\delta,\alpha')$ since it lies in $V(\delta,\alpha')$.

Step 11 (Clause weight). Let $\alpha>0$ and $\varepsilon>0$ be given; the choices are made in the order $(\mu_{0},\nu_{0})$, then $K$, then $\delta_{1}$, and $\delta$ is arbitrary afterwards. By claim 8 of Elementary Order Arithmetic in an Ordered Field, $\varepsilon/2>0$, so claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ gives $(\mu_{0},\nu_{0})\in\mathcal{D}\times\mathcal{D}$ with $\Psi_{0,\alpha}(\mu_{0},\nu_{0})>S(0,\alpha)-\varepsilon/2$. Put $K=|\mathcal{E}(\mu_{0})|+|\mathcal{E}(\nu_{0})|+1$, which is positive: $0\le|\mathcal{E}(\mu_{0})|+|\mathcal{E}(\nu_{0})|$ by claim 1 of Properties of the Absolute Value in an Ordered Field and claim 2 of Elementary Arithmetic in an Ordered Field, so $1=0+1\le K$ by the compatibility of the order with addition in Ordered Field, and $0<1$ by claim 6 of Elementary Order Arithmetic in an Ordered Field, whence $0<K$ by claim 2 there; and $\delta_{1}=\tfrac{\varepsilon}{2}K^{-1}$, which is positive by claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field. Let $0<\delta\le\delta_{1}$. By claim 3 of Properties of the Absolute Value in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field,

$$\delta\bigl(\mathcal{E}(\mu_{0})+\mathcal{E}(\nu_{0})\bigr)\le\delta K\le\delta_{1}K=\tfrac{\varepsilon}{2}.$$

Therefore, since $\Psi_{\delta,\alpha}(\mu_{0},\nu_{0})\in V(\delta,\alpha)$,

$$S(\delta,\alpha)\ge\Psi_{\delta,\alpha}(\mu_{0},\nu_{0})=\Psi_{0,\alpha}(\mu_{0},\nu_{0})-\delta\bigl(\mathcal{E}(\mu_{0})+\mathcal{E}(\nu_{0})\bigr)>S(0,\alpha)-\tfrac{\varepsilon}{2}-\tfrac{\varepsilon}{2}=S(0,\alpha)-\varepsilon. \qquad\blacksquare$$