Proof of Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs

Each result cited is universally quantified over the data in its own statement. Items stated in the setting First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation are taken at the level stated, in the sense of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §levels, with the notation of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §notation. For $i\in\{1,2\}$, $W_{a,i}$ is a metric on $\mathcal{P}^{a}_{\rho_{i}}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric taken at level $i$; so it is nonnegative, symmetric, vanishes on the diagonal and satisfies the triangle inequality (Metric Space). By Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair taken at level $i$, $\mathcal{D}_{i}\subseteq\mathcal{P}^{a}_{\rho_{i}}$ and $\mathcal{E}_{i}:\mathcal{D}_{i}\to\mathbb{R}$. For $c\in\mathbb{R}$ we write $\mathcal{D}_{i,c}=\{\sigma\in\mathcal{D}_{i}:\mathcal{E}_{i}(\sigma)\le c\}$, which is the set written $\mathcal{D}_{c}$ in Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances taken at level $i$ for the pair $\mathcal{P}_{i}$. Suprema and upper bounds are those of The Real Numbers: Standing Notation and Background §bounds. The letter $p$ denotes the mode restriction and $X_{i}$ the Hilbert spaces of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §levels; since $p$ is the mode restriction, and $q$ is reserved for noise fields by Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §notation, the doubling term is called $\vartheta$, and the complete metric space built below is called $Z$.

Preliminaries. Since $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$ are compatible, they are compatible with some nonnegative constant in the sense of Compatible Noise Penalty Pairs at a Fine and a Coarse Level §compatible, whose first condition, together with $\mathcal{D}_{2}\subseteq\mathcal{P}^{a}_{\rho_{2}}$, gives

$$p_{\#}\mu\in\mathcal{D}_{2}\subseteq\mathcal{P}^{a}_{\rho_{2}}\qquad(\mu\in\mathcal{D}_{1}).\qquad(0.1)$$

Let $\mu,\mu'\in\mathcal{D}_{1}$. Both belong to $\mathcal{P}^{a}_{\rho_{1}}$, to any two members of which Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields §lipschitz applies; that claim, applied with $\mu$ and $\mu'$ in the roles of its two measures $\mu$ and $\nu$, gives

$$W_{a,2}(p_{\#}\mu,p_{\#}\mu')\le W_{a,1}(\mu,\mu')\qquad(\mu,\mu'\in\mathcal{D}_{1}).\qquad(0.2)$$

Finally, for $i\in\{1,2\}$, $\delta e_{i}\le\delta\,\mathcal{E}_{i}(\sigma)$ for every $\sigma\in\mathcal{D}_{i}$, by the choice of $e_{i}$ and the positivity of $\delta$.

Clause bounds. Let $\alpha$ be positive and $(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$. Since $\tfrac{\alpha}{2}W_{a,2}(p_{\#}\mu,\nu)^{2}\ge0$, the hypotheses on $U$ and $V$ give $\Psi_{\alpha}(\mu,\nu)\le b_{1}-\delta\,\mathcal{E}_{1}(\mu)+b_{2}-\delta\,\mathcal{E}_{2}(\nu)$, and the preliminaries give $-\delta\bigl(\mathcal{E}_{1}(\mu)+\mathcal{E}_{2}(\nu)\bigr)\le-\delta(e_{1}+e_{2})$; this is the displayed chain of clause bounds. The set of values of $\Psi_{\alpha}$ is nonempty because $\mathcal{D}_{1}\times\mathcal{D}_{2}$ is (as noted in the statement), and it is bounded above by $b_{1}+b_{2}-\delta(e_{1}+e_{2})$, so its supremum $M(\alpha)$ is a real number by The Real Numbers: Standing Notation and Background §bounds. As $M(\alpha)$ is an upper bound of these values,

$$\Psi_{\alpha}(\mu,\nu)\le M(\alpha)\qquad\text{for every positive }\alpha\text{ and all }(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}.\qquad(1.1)$$

Clause diagonal. Let $\alpha$ be positive and $\mu\in\mathcal{D}_{1}$. By (0.1), $p_{\#}\mu\in\mathcal{D}_{2}$, so $(\mu,p_{\#}\mu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$. Since $W_{a,2}(p_{\#}\mu,p_{\#}\mu)=0$, we have $\Psi_{\alpha}(\mu,p_{\#}\mu)=U(\mu)+V(p_{\#}\mu)$, and (1.1) gives $U(\mu)+V(p_{\#}\mu)\le M(\alpha)$.

Clause weight. Let $s$, $U'$, $V'$, $\alpha$, $M'(\alpha)$, $\tau$ and $(\mu,\nu)$ be as in the clause. Adding the two hypotheses on $U'$ and $V'$, taken at $\mu$ and at $\nu$, and subtracting $\tfrac{\alpha}{2}W_{a,2}(p_{\#}\mu,\nu)^{2}$,

$$\Psi_{\alpha}(\mu,\nu)+s\bigl(\mathcal{E}_{1}(\mu)+\mathcal{E}_{2}(\nu)\bigr)\le U'(\mu)+V'(\nu)-\tfrac{\alpha}{2}W_{a,2}(p_{\#}\mu,\nu)^{2}=\Psi'_{\alpha}(\mu,\nu).$$

As $M'(\alpha)$ is the supremum of the values of $\Psi'_{\alpha}$, it is an upper bound of them; together with the hypothesis $M(\alpha)-\tau\le\Psi_{\alpha}(\mu,\nu)$ this gives

$$M(\alpha)-\tau+s\bigl(\mathcal{E}_{1}(\mu)+\mathcal{E}_{2}(\nu)\bigr)\le\Psi_{\alpha}(\mu,\nu)+s\bigl(\mathcal{E}_{1}(\mu)+\mathcal{E}_{2}(\nu)\bigr)\le\Psi'_{\alpha}(\mu,\nu)\le M'(\alpha).$$

Clause strength. Let $0<\alpha'<\alpha$, $0\le\tau$ and let $(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ satisfy $M(\alpha)-\tau\le\Psi_{\alpha}(\mu,\nu)$. Since $\tfrac{\alpha'}{2}=\tfrac{\alpha}{2}-\tfrac{\alpha-\alpha'}{2}$, directly from the definition

$$\Psi_{\alpha'}(\mu,\nu)=\Psi_{\alpha}(\mu,\nu)+\tfrac{\alpha-\alpha'}{2}W_{a,2}(p_{\#}\mu,\nu)^{2}\qquad\text{for all }(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}.\qquad(4.1)$$

Adding $\tfrac{\alpha-\alpha'}{2}W_{a,2}(p_{\#}\mu,\nu)^{2}$ to both sides of the hypothesis and using (1.1) for $\alpha'$,

$$M(\alpha)-\tau+\tfrac{\alpha-\alpha'}{2}W_{a,2}(p_{\#}\mu,\nu)^{2}\le\Psi_{\alpha'}(\mu,\nu)\le M(\alpha').$$

Clause monotone. Let $0<\alpha'<\alpha$. For every $(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ the term $\tfrac{\alpha-\alpha'}{2}W_{a,2}(p_{\#}\mu,\nu)^{2}$ is nonnegative, so (4.1) and (1.1) for $\alpha'$ give $\Psi_{\alpha}(\mu,\nu)\le\Psi_{\alpha'}(\mu,\nu)\le M(\alpha')$. Thus $M(\alpha')$ is an upper bound of the values of $\Psi_{\alpha}$, and since $M(\alpha)$ is their least upper bound, $M(\alpha)\le M(\alpha')$.

Clause perturbed, part 1: the number $m$ and the constants of clause radius. Let $m$ be as in the clause. For every $\mu_{0}\in\mathcal{D}_{1}$, clause diagonal gives $U(\mu_{0})+V(p_{\#}\mu_{0})\le M(\alpha)$ for every positive $\alpha$, so this number is an admissible $m$, as the clause asserts. The constants are chosen in the order $K$, then $B_{0,1}$ and $B_{0,2}$, then $B_{1}$ and $B_{2}$, all before $\alpha$ and $\tau$. Put

$$K=\delta^{-1}(b_{1}+b_{2}-m+1)+|e_{1}|+|e_{2}|,$$

which depends only on $\delta$, $b_{1}$, $b_{2}$, $e_{1}$, $e_{2}$ and $m$. For $i\in\{1,2\}$ let $B_{0,i}\in\mathbb{R}$ satisfy $W_{a,i}(\sigma,\rho_{i})\le B_{0,i}$ for every $\sigma\in\mathcal{D}_{i,K}$, as provided by Noise-Closed Noise Penalty Pairs §bounded taken at level $i$ with $c=K$ (the pair $\mathcal{P}_{i}$ being noise-closed, Noise-Closed Noise Penalty Pairs §noise-closed), and put $B_{i}=|B_{0,i}|$. Then $0\le B_{i}$, $B_{i}$ depends only on $K$ and $\mathcal{P}_{i}$, none of $K$, $B_{1}$, $B_{2}$ depends on $\alpha$ or $\tau$, and $W_{a,i}(\sigma,\rho_{i})\le B_{0,i}\le B_{i}$ for every $\sigma\in\mathcal{D}_{i}$ with $\mathcal{E}_{i}(\sigma)\le K$, which is the display of clause radius. By Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §diameter taken at level $i$, with $c=K$ and the bound $B_{i}$,

$$W_{a,i}(\sigma,\sigma')\le2B_{i}\qquad\text{and hence}\qquad W_{a,i}(\sigma,\sigma')^{2}\le4B_{i}^{2}\qquad(\sigma,\sigma'\in\mathcal{D}_{i,K},\ i\in\{1,2\}),\qquad(5.1)$$

the second by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both $W_{a,i}(\sigma,\sigma')$ and $2B_{i}$ being nonnegative.

Off the localising set. Let $\alpha$ be positive and let $(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ with $\mu\notin\mathcal{D}_{1,K}$ or $\nu\notin\mathcal{D}_{2,K}$. Then

$$\Psi_{\alpha}(\mu,\nu)<m-1\le M(\alpha)-1.\qquad(5.2)$$

Indeed, $\delta K=b_{1}+b_{2}-m+1+\delta(|e_{1}|+|e_{2}|)$, and since $0\le|e_{1}|$, $0\le|e_{2}|$, $0\le e_{1}+|e_{1}|$, $0\le e_{2}+|e_{2}|$ and $\delta$ is positive, both $\delta K+\delta e_{1}$ and $\delta K+\delta e_{2}$ are at least $b_{1}+b_{2}-m+1$. If $\mu\notin\mathcal{D}_{1,K}$, then $K<\mathcal{E}_{1}(\mu)$, and clause bounds together with $\delta e_{2}\le\delta\,\mathcal{E}_{2}(\nu)$ (preliminaries) and the positivity of $\delta$ gives

$$\Psi_{\alpha}(\mu,\nu)\le b_{1}+b_{2}-\delta\,\mathcal{E}_{1}(\mu)-\delta e_{2}<b_{1}+b_{2}-\delta K-\delta e_{2}\le m-1;$$

if $\nu\notin\mathcal{D}_{2,K}$, the same holds with the roles of the two levels exchanged, using $\delta e_{1}\le\delta\,\mathcal{E}_{1}(\mu)$ and $K<\mathcal{E}_{2}(\nu)$. The last inequality of (5.2) is the hypothesis $m\le M(\alpha)$.

Clause perturbed, part 2: the complete metric space $Z$. Let $d_{\times}$ be the product metric on $\mathcal{P}^{a}_{\rho_{1}}\times\mathcal{P}^{a}_{\rho_{2}}$ built from the metric spaces $(\mathcal{P}^{a}_{\rho_{1}},W_{a,1})$ and $(\mathcal{P}^{a}_{\rho_{2}},W_{a,2})$, a metric by claim 1 of The Product Metric is a Metric. By claim 2 of that theorem, for $z=(\mu,\nu)$ and $z'=(\mu',\nu')$ we have $W_{a,1}(\mu,\mu')\le d_{\times}(z,z')$ and $W_{a,2}(\nu,\nu')\le d_{\times}(z,z')$. Put $Z=\mathcal{D}_{1,K}\times\mathcal{D}_{2,K}\subseteq\mathcal{D}_{1}\times\mathcal{D}_{2}$ and let $d_{Z}$ be the restriction of $d_{\times}$ to $Z\times Z$, a metric on $Z$ by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology.

$(Z,d_{Z})$ is complete. Let $(z_{n})_{n\in\mathbb{N}}$ be a Cauchy sequence in $(Z,d_{Z})$, $z_{n}=(\sigma_{n},\sigma'_{n})$. Given a positive $\varepsilon$ and $N$ as in that definition, the domination above gives $W_{a,1}(\sigma_{n},\sigma_{l})\le d_{Z}(z_{n},z_{l})<\varepsilon$ and $W_{a,2}(\sigma'_{n},\sigma'_{l})<\varepsilon$ for all $n,l\ge N$; so $(\sigma_{n})$ is a Cauchy sequence in $(\mathcal{D}_{1,K},W_{a,1})$ and $(\sigma'_{n})$ one in $(\mathcal{D}_{2,K},W_{a,2})$, and by Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §complete, taken at level 1 and at level 2 with $c=K$, they converge in these spaces to points $\sigma\in\mathcal{D}_{1,K}$ and $\sigma'\in\mathcal{D}_{2,K}$. Since that convergence is expressed by the same distances $W_{a,1}(\sigma_{n},\sigma)$ and $W_{a,2}(\sigma'_{n},\sigma')$, the two sequences also converge in $(\mathcal{P}^{a}_{\rho_{1}},W_{a,1})$ and in $(\mathcal{P}^{a}_{\rho_{2}},W_{a,2})$, and by claim 1 of Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space $(z_{n})$ converges to $(\sigma,\sigma')$ in $(\mathcal{P}^{a}_{\rho_{1}}\times\mathcal{P}^{a}_{\rho_{2}},d_{\times})$. As $(\sigma,\sigma')\in Z$ and $d_{Z}$ is the restriction of $d_{\times}$, $(z_{n})$ converges to $(\sigma,\sigma')$ in $(Z,d_{Z})$.

Clause perturbed, part 3: two elementary facts. (a) If $s,s',t$ are nonnegative reals with $s\le s'+t$, then $s^{2}\le s'^{2}+2st$: if $s\le s'$ then $s^{2}\le s'^{2}$; otherwise $0<s-s'\le t$ and $s+s'\le2s$, so $s^{2}-s'^{2}=(s-s')(s+s')\le2st$. (b) Let $(Y,d_{Y})$ be a metric space, $A\subseteq Y$, $f_{0}:A\to\mathbb{R}$, $y\in A$, and suppose there is a nonnegative $\ell$ with $f_{0}(y)\le f_{0}(y')+\ell\,d_{Y}(y,y')$ for every $y'\in A$. Then $f_{0}$ is lower semicontinuous at $y$ relative to $A$: given a positive $\varepsilon$, the number $r=\varepsilon(\ell+1)^{-1}$ is positive, and for $y'\in A$ with $d_{Y}(y,y')<r$ we get $\ell\,d_{Y}(y,y')\le\ell r<\varepsilon$, hence $f_{0}(y)-\varepsilon<f_{0}(y')$.

Clause perturbed, part 4: the function $f$. Let $\alpha$ be positive and write $\Psi=\Psi_{\alpha}$, $M=M(\alpha)$. (a) By claim 1 of Negation, Restriction, and Separated Differences of Semicontinuous Functions, applied at each point of $\mathcal{D}_{2}$ in the metric space $(\mathcal{P}^{a}_{\rho_{2}},W_{a,2})$ with the subset $\mathcal{D}_{2}$, the function $-V$ is lower semicontinuous on $\mathcal{D}_{2}$ relative to $\mathcal{D}_{2}$, since $V$ is upper semicontinuous there. By claim 4 of that lemma, with its two metric spaces $(\mathcal{P}^{a}_{\rho_{1}},W_{a,1})$ and $(\mathcal{P}^{a}_{\rho_{2}},W_{a,2})$, its two subsets $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$, its upper semicontinuous function $U$ and its lower semicontinuous function $-V$, the function $h(\mu,\nu)=U(\mu)-(-V(\nu))=U(\mu)+V(\nu)$ is upper semicontinuous on $\mathcal{D}_{1}\times\mathcal{D}_{2}$ with respect to $d_{\times}$. (b) Let $\vartheta(\mu,\nu)=\tfrac{\alpha}{2}W_{a,2}(p_{\#}\mu,\nu)^{2}$ on $\mathcal{D}_{1}\times\mathcal{D}_{2}$. Let $z=(\mu,\nu)$ and $z'=(\mu',\nu')$ lie in $\mathcal{D}_{1}\times\mathcal{D}_{2}$, put $t=d_{\times}(z,z')$, $s=W_{a,2}(p_{\#}\mu,\nu)$ and $s'=W_{a,2}(p_{\#}\mu',\nu')$. By (0.1) the measures $p_{\#}\mu$, $p_{\#}\mu'$, $\nu'$ and $\nu$ lie in $\mathcal{P}^{a}_{\rho_{2}}$, so the triangle inequality and symmetry of $W_{a,2}$, then (0.2) and the domination of part 2, give

$$s\le W_{a,2}(p_{\#}\mu,p_{\#}\mu')+W_{a,2}(p_{\#}\mu',\nu')+W_{a,2}(\nu',\nu)\le W_{a,1}(\mu,\mu')+s'+W_{a,2}(\nu,\nu')\le s'+2t.$$

Part 3(a), with $2t$ in place of $t$, gives $s^{2}\le s'^{2}+4ts$, so $\vartheta(z)\le\vartheta(z')+2\alpha s\,t$. By part 3(b), with $\ell=2\alpha s$, $\vartheta$ is lower semicontinuous at every point of $\mathcal{D}_{1}\times\mathcal{D}_{2}$ relative to $\mathcal{D}_{1}\times\mathcal{D}_{2}$ in $(\mathcal{P}^{a}_{\rho_{1}}\times\mathcal{P}^{a}_{\rho_{2}},d_{\times})$. (c) Hence $\Psi=h-\vartheta$ is upper semicontinuous at every point of $\mathcal{D}_{1}\times\mathcal{D}_{2}$ by claim 3 of Negation, Restriction, and Separated Differences of Semicontinuous Functions, and its restriction $f:Z\to\mathbb{R}$, $f(z)=\Psi(z)$, is upper semicontinuous at every point of $Z$ relative to $Z$ by claim 2 of that lemma. Since $d_{Z}$ agrees with $d_{\times}$ on $Z\times Z$, the condition of Upper Semicontinuous Function on a Subset of a Metric Space reads the same in $(Z,d_{Z})$, so $f$ is upper semicontinuous on $Z$ in $(Z,d_{Z})$. By clause bounds, $f$ is bounded above by $b_{1}+b_{2}-\delta(e_{1}+e_{2})$.

Clause perturbed, part 5: the weights and the starting point. Let $\alpha,\tau\in\mathbb{R}$ with $0<\alpha$ and $0<\tau<1$, with $\Psi$, $M$ and $f$ as in part 4; they are chosen after $K$, $B_{1}$ and $B_{2}$. (a) Put $c_{k}=\tau\bigl(\tfrac12\bigr)^{k}$ for $k\in\mathbb{N}$, a positive real. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, $\sum_{k=1}^{\infty}(\tfrac12)^{k}$ converges with sum $1$, so by Elementary Properties of Series of Real Numbers §linearity (both of its series taken to be this one, and $\lambda=\tau$) the series $\sum_{k=1}^{\infty}c_{k}$ converges with sum $\tau$; in particular $\sum_{k=1}^{\infty}c_{k}\le\tau$. (b) By Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ §epsilon-above, applied to the nonempty set of values of $\Psi$, bounded above with supremum $M$ (clause bounds), and to $\varepsilon=\tau$, there is $z_{1}=(\mu_{1},\nu_{1})\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ with $M-\tau<\Psi(z_{1})$. As $\tau<1$, $M-1<M-\tau$, so by (5.2) neither $\mu_{1}\notin\mathcal{D}_{1,K}$ nor $\nu_{1}\notin\mathcal{D}_{2,K}$ can hold; thus $z_{1}\in Z$ and $Z\ne\varnothing$. The values of $f$ are values of $\Psi$, so $f$ is bounded above by $M$ and $\sup_{z\in Z}f(z)\le M$; hence $f(z_{1})>M-\tau\ge\sup_{z\in Z}f(z)-\tau$.

Clause perturbed, part 6: the gauge. For $z=(\mu,\nu)$ and $y=(\mu',\nu')$ in $Z$ put $g(z,y)=W_{a,1}(\mu,\mu')^{2}+W_{a,2}(\nu,\nu')^{2}$, and $G=4B_{1}^{2}+4B_{2}^{2}$, a nonnegative real. (i) $g(z,z)=0$ and $0\le g(z,y)$. Since $\mu,\mu'\in\mathcal{D}_{1,K}$ and $\nu,\nu'\in\mathcal{D}_{2,K}$, (5.1) gives $W_{a,1}(\mu,\mu')^{2}\le4B_{1}^{2}$ and $W_{a,2}(\nu,\nu')^{2}\le4B_{2}^{2}$, so $g(z,y)\le G$. (ii) Fix $y=(\mu'',\nu'')\in Z$. For $z=(\mu,\nu)$, $z'=(\mu',\nu')$ in $Z$ and $t=d_{Z}(z,z')$, the triangle inequalities of $W_{a,1}$ and $W_{a,2}$ and the domination of part 2 give $W_{a,1}(\mu,\mu'')\le W_{a,1}(\mu',\mu'')+t$ and $W_{a,2}(\nu,\nu'')\le W_{a,2}(\nu',\nu'')+t$, so part 3(a) gives $g(z,y)\le g(z',y)+\ell t$ with $\ell=2\bigl(W_{a,1}(\mu,\mu'')+W_{a,2}(\nu,\nu'')\bigr)\ge0$; by part 3(b), applied in $(Z,d_{Z})$ with $A=Z$, $g(\cdot,y)$ is lower semicontinuous on $Z$. (iii) Let $\eta$ be positive and $\beta=(\tfrac{\eta}{2})^{2}$, which is positive. If $z=(\mu,\nu)$ and $y=(\mu',\nu')$ in $Z$ satisfy $g(z,y)\le\beta$, then $W_{a,1}(\mu,\mu')^{2}\le\beta$, so $W_{a,1}(\mu,\mu')\le\tfrac{\eta}{2}<\eta$ (if $\tfrac{\eta}{2}<W_{a,1}(\mu,\mu')$ then $\beta<W_{a,1}(\mu,\mu')^{2}$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field); likewise $W_{a,2}(\nu,\nu')<\eta$, and $d_{Z}(z,y)<\eta$ by claim 3 of The Product Metric is a Metric.

Clause perturbed, part 7: the variational principle, and clauses localised, near-maximiser and strict-maximum. With $\alpha,\tau$ as in part 5, apply A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space, with $(Z,d_{Z})$ in place of its $(X,d)$, to this nonempty complete metric space (parts 2 and 5(b)), the upper semicontinuous function $f$, bounded above (part 4), the bound $G$ and the gauge $g$ (part 6), the weights $c_{k}$ (part 5(a)), its $\varepsilon$ taken to be $\tau$, and the point $z_{1}$ (part 5(b)). It yields $(\hat{\mu},\hat{\nu})\in Z$ in place of its $\bar{x}$ and a sequence $(z_{k})_{k\in\mathbb{N}}$ in $Z$ with first term $z_{1}$; write $z_{k}=(\mu_{k},\nu_{k})$, so that $(\mu_{k})$ is a sequence in $\mathcal{D}_{1,K}\subseteq\mathcal{D}_{1}$ and $(\nu_{k})$ one in $\mathcal{D}_{2,K}\subseteq\mathcal{D}_{2}$.

Clause localised. Since $(\hat{\mu},\hat{\nu})$ and every $z_{k}$ lie in $Z=\mathcal{D}_{1,K}\times\mathcal{D}_{2,K}$, we have $\mathcal{E}_{1}(\hat{\mu})\le K$, $\mathcal{E}_{2}(\hat{\nu})\le K$, $\mathcal{E}_{1}(\mu_{k})\le K$ and $\mathcal{E}_{2}(\nu_{k})\le K$ for every $k$.

Convergence of the series. Let $(\mu,\nu)\in\mathcal{P}^{a}_{\rho_{1}}\times\mathcal{P}^{a}_{\rho_{2}}$ and $w_{k}=W_{a,1}(\mu,\mu_{k})^{2}+W_{a,2}(\nu,\nu_{k})^{2}$. For each $k$, $W_{a,1}(\mu,\mu_{k})\le W_{a,1}(\mu,\mu_{1})+W_{a,1}(\mu_{1},\mu_{k})$, and $W_{a,1}(\mu_{1},\mu_{k})\le2B_{1}$ by (5.1) ($\mu_{1},\mu_{k}\in\mathcal{D}_{1,K}$); since $(s+t)^{2}\le2s^{2}+2t^{2}$ for reals $s,t$ and squares are monotone on nonnegative reals (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), this gives $W_{a,1}(\mu,\mu_{k})^{2}\le2W_{a,1}(\mu,\mu_{1})^{2}+8B_{1}^{2}$, and in the same way, with $\nu_{1},\nu_{k}\in\mathcal{D}_{2,K}$, $W_{a,2}(\nu,\nu_{k})^{2}\le2W_{a,2}(\nu,\nu_{1})^{2}+8B_{2}^{2}$. So $0\le w_{k}\le L$ with $L=2W_{a,1}(\mu,\mu_{1})^{2}+2W_{a,2}(\nu,\nu_{1})^{2}+8B_{1}^{2}+8B_{2}^{2}$, independent of $k$ and nonnegative. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §tail-bound, with its nonnegative convergent series $\sum_{k=1}^{\infty}c_{k}$ and its bound $L$, the series $\sum_{k=1}^{\infty}c_{k}w_{k}$ converges. Its terms are nonnegative, so its sum is at least its first partial sum, which is nonnegative, by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates. Hence, for $\Phi$ as in clause strict-maximum,

$$\Phi(\mu,\nu)\le\Psi(\mu,\nu)\qquad\text{for every }(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}.\qquad(7.1)$$

For $z\in Z$ the sum of this series is $\sum_{k=1}^{\infty}c_{k}\,g(z,z_{k})$, so the function $\Phi$ of A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space, formed from $f$, $g$, the $c_{k}$ and the $z_{k}$, is the restriction of the present $\Phi$ to $Z$.

Clause near-maximiser. By A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space §value and part 5(b), $\Phi(\hat{\mu},\hat{\nu})\ge f(z_{1})>M-\tau$, so by (7.1) $\Psi_{\alpha}(\hat{\mu},\hat{\nu})\ge\Phi(\hat{\mu},\hat{\nu})>M(\alpha)-\tau$.

Clause strict-maximum. Let $(\mu,\nu)\in\mathcal{D}_{1}\times\mathcal{D}_{2}$ with $(\mu,\nu)\ne(\hat{\mu},\hat{\nu})$. If $(\mu,\nu)\in Z$, then $\Phi(\mu,\nu)<\Phi(\hat{\mu},\hat{\nu})$ by A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space §maximum. Otherwise $\mu\notin\mathcal{D}_{1,K}$ or $\nu\notin\mathcal{D}_{2,K}$, and (7.1), (5.2), $\tau<1$ and the inequality $\Phi(\hat{\mu},\hat{\nu})>M-\tau$ established above give

$$\Phi(\mu,\nu)\le\Psi(\mu,\nu)<M-1<M-\tau<\Phi(\hat{\mu},\hat{\nu}).$$

Since $K$, $B_{1}$ and $B_{2}$ were fixed in part 1 before $\alpha$ and $\tau$, clause perturbed holds. $\blacksquare$