Proof of The Distance between Square-Integrable Noncommutative Laws is the Infimum of the Cost over Their Couplings

Each result cited below is universally quantified over the data in its own statement.

Conventions. For $k\in\mathbb{N}$, $(\Sigma_{k},W_{2})$ is a metric space by The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §metric, and $\Sigma^{2}_{k}$ is its metric completion with metric $\widehat{W}_{2}$ and canonical map $\kappa_{k}$ (Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws), a metric space by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §metric. $\mathrm{pr}^{1},\mathrm{pr}^{2},D$ are the coordinate data and $\mathrm{pr}^{\epsilon}_{\#}$, $D_{\#}$ their push-forwards; $\iota^{1},\iota^{2}$, $\Pi$, $I$ and $\Delta$ are as in Couplings of Two Noncommutative Laws and Their Quadratic Cost, applied with $d$ or with $2d$ variables as indicated. Every element of $\Sigma_{k}$ is a tracial state lying in some $\Sigma_{k,R}$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law), so by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone finitely many elements of $\Sigma_{k}$ lie in one common $\Sigma_{k,R}$ (take $R$ the largest of their bounds). Costs of couplings are real and nonnegative by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §cost, and all square roots are the nonnegative ones of Existence and Uniqueness of the Nonnegative Square Root. For real $s,t\ge0$, $s^{2}\le t^{2}$ holds if and only if $s\le t$ (if $s\le t$ then $s^{2}\le st\le t^{2}$; if $t<s$ then $s>0$ and $t^{2}\le ts<s^{2}$), so inequalities between nonnegative reals may be squared or square-rooted. Since $d_{\mathbb{R}}(a,L)=|a-L|$ for the metric of The Absolute Value Metric on the Real Line, convergence in $(\mathbb{R},d_{\mathbb{R}})$ in the sense of Convergent Sequence in a Metric Space is convergence in the sense of Limit of a Sequence of Real Numbers.

Step 1 (a gluing estimate). Claim: let $R>0$ be real, $\lambda_{1},\lambda_{2},\lambda_{3}\in\Sigma_{d,R}$, $\gamma_{1}\in\Pi(\lambda_{1},\lambda_{2})$ and $\gamma_{2}\in\Pi(\lambda_{2},\lambda_{3})$. Then $\gamma_{1},\gamma_{2}\in\Sigma_{2d}$, and there is $\gamma_{3}\in\Pi(\lambda_{1},\lambda_{3})\cap\Sigma_{2d}$ with

where $W_{2}$ is the distance on $\Sigma_{2d}$.

Apply Gluing Two Noncommutative Couplings along a Common Marginal with its laws $\mu,\nu,\rho$ taken to be $\lambda_{1},\lambda_{2},\lambda_{3}$ and its couplings $\gamma_{1},\gamma_{2}$ taken to be ours: it gives $\psi\in\Sigma_{3d,R}$ with $\psi\circ\sigma^{12}=\gamma_{1}$ (Gluing Two Noncommutative Couplings along a Common Marginal §first), $\psi\circ\sigma^{23}=\gamma_{2}$ (Gluing Two Noncommutative Couplings along a Common Marginal §second) and $\gamma_{3}:=\psi\circ\sigma^{13}\in\Pi(\lambda_{1},\lambda_{3})$ (Gluing Two Noncommutative Couplings along a Common Marginal §composite), where $\sigma^{12},\sigma^{23},\sigma^{13}$ are the substitutions named there. By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound, $\gamma_{1},\gamma_{2},\gamma_{3}\in\Sigma_{2d,R}\subseteq\Sigma_{2d}$. In $\mathcal{P}_{3d}$ write $u_{j}=x_{j}$, $v_{j}=x_{d+j}$ and $w_{j}=x_{2d+j}$ for $j\in[d]$, self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint; thus $\sigma^{12}$, $\sigma^{23}$ and $\sigma^{13}$ are the substitutions of $(u,v)$, $(v,w)$ and $(u,w)$, where $(u,v)$ denotes $(u_{1},\dots,u_{d},v_{1},\dots,v_{d})$. Let $b=(u,v,u,w)$ and $b'=(v,w,u,w)$, $4d$-tuples in $\mathcal{P}_{3d,\mathrm{sa}}$ formed in the same way, and put $\theta=\psi\circ\sigma_{b}$ and $\theta'=\psi\circ\sigma_{b'}$; they lie in $\Sigma_{4d}$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §law, so they are tracial states on $\mathcal{P}_{4d}$. Let $\iota^{1},\iota^{2}:\mathcal{P}_{2d}\to\mathcal{P}_{4d}$ be the marginal substitutions for $2d$ variables (Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals). By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, $\sigma_{b}\circ\iota^{1}$ is the substitution of $(b_{1},\dots,b_{2d})=(u,v)$, that is $\sigma^{12}$, and $\sigma_{b}\circ\iota^{2}$ is that of $(b_{2d+1},\dots,b_{4d})=(u,w)$, that is $\sigma^{13}$; likewise $\sigma_{b'}\circ\iota^{1}=\sigma^{23}$ and $\sigma_{b'}\circ\iota^{2}=\sigma^{13}$. Hence $\theta\circ\iota^{1}=\gamma_{1}$, $\theta\circ\iota^{2}=\gamma_{3}$, $\theta'\circ\iota^{1}=\gamma_{2}$ and $\theta'\circ\iota^{2}=\gamma_{3}$, that is $\theta\in\Pi(\gamma_{1},\gamma_{3})$ and $\theta'\in\Pi(\gamma_{2},\gamma_{3})$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling (for $2d$ variables).

Substitutions are linear (Substitution of Noncommutative Polynomials into the Variables §substitution), multiplicative (Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism) and send $x_{i}$ to the $i$-th entry of the tuple (Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values). Splitting the sum defining $\Delta_{2d}$ at $i=d$ (claim 1 of Properties of Finite Sums) and using $(-q)^{2}=q^{2}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra),

By Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost, $I(\theta)=\psi(\sigma^{23}(\Delta_{d}))=\gamma_{2}(\Delta_{d})=I(\gamma_{2})$ and $I(\theta')=I(\gamma_{1})$. By The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance (for $2d$ variables), $W_{2}(\gamma_{1},\gamma_{3})^{2}\le I(\theta)=I(\gamma_{2})$ and $W_{2}(\gamma_{2},\gamma_{3})^{2}\le I(\theta')=I(\gamma_{1})$, which give the first two inequalities.

For the third, put $p_{j}=u_{j}-v_{j}$ and $q_{j}=v_{j}-w_{j}$ in $\mathcal{P}_{3d,\mathrm{sa}}$, let $\beta$ be the pairing $\beta_{\psi}$ of Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §pairing, and let $\|\cdot\|_{\beta}$ be its seminorm as in Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences, with $P_{j}=\|p_{j}\|_{\beta}$ and $Q_{j}=\|q_{j}\|_{\beta}$. As above, $\sigma^{12}(\Delta_{d})=\sum_{j}p_{j}^{2}$, $\sigma^{23}(\Delta_{d})=\sum_{j}q_{j}^{2}$ and $\sigma^{13}(\Delta_{d})=\sum_{j}(p_{j}+q_{j})^{2}$, so by linearity of $\psi$ and the symmetry and bilinearity of $\beta$,

By Cauchy Sequences for a Positive Semidefinite Symmetric Bilinear Form: Cauchy-Schwarz, the Space of Cauchy Sequences, Convergence of Pairings, and Null Sequences §cauchy-schwarz, $\beta(p_{j},q_{j})\le P_{j}Q_{j}$; and by Cauchy-Schwarz Inequality for the Euclidean Dot Product for the points $(P_{j})_{j}$ and $(Q_{j})_{j}$ of $\mathbb{R}^{d}$, whose dot product and Euclidean norms are $\sum_{j}P_{j}Q_{j}$, $I(\gamma_{1})^{1/2}$ and $I(\gamma_{2})^{1/2}$ (Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$, Euclidean Norm on $\mathbb{R}^n$), $\sum_{j}P_{j}Q_{j}\le I(\gamma_{1})^{1/2}I(\gamma_{2})^{1/2}$. Hence $I(\gamma_{3})\le(I(\gamma_{1})^{1/2}+I(\gamma_{2})^{1/2})^{2}$, and the third inequality follows by taking square roots. This proves the claim.

Proof of clause 1 (Almost optimal couplings). Fix a real $\varepsilon>0$. Put $w=\widehat{W}_{2}(\mu,\nu)\ge0$ and $\eta=\min\{1,\varepsilon/(2w+1)\}>0$. Then $\eta^{2}\le\eta$ and $(w+\eta)^{2}=w^{2}+2w\eta+\eta^{2}\le w^{2}+\eta(2w+1)\le w^{2}+\varepsilon$. Write $h_{k}=(\tfrac12)^{k}$ for $k\ge0$. By Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric, $h_{k}\to0$ and $\sum_{i=1}^{n}h_{i}=1-h_{n}$ for $n\in\mathbb{N}$; hence $\sum_{i=1}^{n}h_{i}\le1$, and $\sum_{i=k}^{l-1}h_{i}=h_{k-1}-h_{l-1}<h_{k-1}=2h_{k}$ for $1\le k<l$. The choices below are made in the order (a), (b), (d); step (c) is a statement proved for all its data.

(a) Approximants. By The Metric Completion of a Metric Space §completion choose Cauchy sequences $x=(x_{l})$ and $y=(y_{l})$ in $(\Sigma_{d},W_{2})$ with $\mu=[x]$ and $\nu=[y]$; then $\kappa_{d}(x_{l})\to\mu$ and $\kappa_{d}(y_{l})\to\nu$ by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §density. For $k\in\mathbb{N}$, the set of $N\in\mathbb{N}$ such that $\widehat{W}_{2}(\kappa_{d}(x_{l}),\mu)<\eta h_{k}/8$ and $\widehat{W}_{2}(\kappa_{d}(y_{l}),\nu)<\eta h_{k}/8$ for all $l\ge N$ is nonempty by Convergent Sequence in a Metric Space (take the larger of the two indices given there); let $N_{k}$ be its least element (The Natural Numbers Are Well Ordered), so that no choice is involved, and put $\mu_{k}=x_{N_{k}}$ and $\nu_{k}=y_{N_{k}}$, elements of $\Sigma_{d}$. Thus $\widehat{W}_{2}(\kappa_{d}(\mu_{k}),\mu)<\eta h_{k}/8$ and $\widehat{W}_{2}(\kappa_{d}(\nu_{k}),\nu)<\eta h_{k}/8$ for every $k$; as $h_{k}\to0$, $\kappa_{d}(\mu_{k})\to\mu$ and $\kappa_{d}(\nu_{k})\to\nu$. Put $a_{k}=W_{2}(\mu_{k},\mu_{k+1})$ and $b_{k}=W_{2}(\nu_{k},\nu_{k+1})$. By The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry and the triangle inequality for $\widehat{W}_{2}$, $a_{k}\le\widehat{W}_{2}(\kappa_{d}(\mu_{k}),\mu)+\widehat{W}_{2}(\mu,\kappa_{d}(\mu_{k+1}))<\eta h_{k}/8+\eta h_{k+1}/8<\eta h_{k}/4$, likewise $b_{k}<\eta h_{k}/4$, and

(b) Start. Take $R>0$ with $\mu_{1},\nu_{1}\in\Sigma_{d,R}$ and choose, by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained, an optimal $\gamma_{1}\in\Pi(\mu_{1},\nu_{1})$ (The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal); then $I(\gamma_{1})^{1/2}=W_{2}(\mu_{1},\nu_{1})<w+\eta/4$.

(c) One step. Claim: for every $k\in\mathbb{N}$ and every $\gamma\in\Pi(\mu_{k},\nu_{k})$ there is $\tilde\gamma\in\Pi(\mu_{k+1},\nu_{k+1})$ with $W_{2}(\gamma,\tilde\gamma)\le a_{k}+b_{k}$ and $I(\tilde\gamma)^{1/2}\le I(\gamma)^{1/2}+a_{k}+b_{k}$. Take $R>0$ with $\mu_{k},\mu_{k+1},\nu_{k},\nu_{k+1}\in\Sigma_{d,R}$. Step A: by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained choose an optimal $\alpha\in\Pi(\mu_{k+1},\mu_{k})$; by The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §symmetry, $I(\alpha)=W_{2}(\mu_{k+1},\mu_{k})^{2}=a_{k}^{2}$, so $I(\alpha)^{1/2}=a_{k}$. Step 1 with $(\lambda_{1},\lambda_{2},\lambda_{3})=(\mu_{k+1},\mu_{k},\nu_{k})$ and $(\gamma_{1},\gamma_{2})=(\alpha,\gamma)$ gives $\gamma'\in\Pi(\mu_{k+1},\nu_{k})$ with $W_{2}(\gamma,\gamma')\le I(\alpha)^{1/2}=a_{k}$ and $I(\gamma')^{1/2}\le a_{k}+I(\gamma)^{1/2}$. Step B: choose an optimal $\beta\in\Pi(\nu_{k},\nu_{k+1})$, so $I(\beta)^{1/2}=b_{k}$. Step 1 with $(\lambda_{1},\lambda_{2},\lambda_{3})=(\mu_{k+1},\nu_{k},\nu_{k+1})$ and $(\gamma_{1},\gamma_{2})=(\gamma',\beta)$ gives $\tilde\gamma\in\Pi(\mu_{k+1},\nu_{k+1})$ with $W_{2}(\gamma',\tilde\gamma)\le I(\beta)^{1/2}=b_{k}$ and $I(\tilde\gamma)^{1/2}\le I(\gamma')^{1/2}+b_{k}$. By The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §triangle in $\Sigma_{2d}$, $W_{2}(\gamma,\tilde\gamma)\le a_{k}+b_{k}$, and $I(\tilde\gamma)^{1/2}\le I(\gamma)^{1/2}+a_{k}+b_{k}$.

(d) Recursion by dependent choice. Each step of the construction chooses among couplings, and no choice function on the sets $\Pi(\mu_{k},\nu_{k})$ is available, so we use Axiom of Dependent Choice rather than Definition of Sequences by Recursion on the Natural Numbers. Let $\mathcal{S}$ be the set of pairs $(k,\gamma)$ with $k\in\mathbb{N}$ and $\gamma\in\Pi(\mu_{k},\nu_{k})$, and let $\mathcal{R}$ be the relation on $\mathcal{S}$ consisting of the pairs $((k,\gamma),(k',\tilde\gamma))$ with $k'=k+1$, $W_{2}(\gamma,\tilde\gamma)\le a_{k}+b_{k}$ and $I(\tilde\gamma)^{1/2}\le I(\gamma)^{1/2}+a_{k}+b_{k}$. By (c) every element of $\mathcal{S}$ has an $\mathcal{R}$-successor, and $(1,\gamma_{1})\in\mathcal{S}$. The axiom gives a sequence $(s_{k})_{k\in\mathbb{N}}$ in $\mathcal{S}$ with $s_{1}=(1,\gamma_{1})$ and $(s_{k},s_{k+1})\in\mathcal{R}$ for all $k$; by induction the first entry of $s_{k}$ is $k$, so $s_{k}=(k,\gamma_{k})$ with $\gamma_{k}\in\Pi(\mu_{k},\nu_{k})$, and

(e) Cauchy property and cost bound. Each $\gamma_{k}$ lies in $\Sigma_{2d}$ by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound. For $1\le k<l$, induction on $l$ with The Noncommutative Wasserstein Distance Satisfies the Triangle Inequality and is a Metric on Noncommutative Laws §triangle gives $W_{2}(\gamma_{k},\gamma_{l})\le\sum_{i=k}^{l-1}(a_{i}+b_{i})<\frac{\eta}{2}\sum_{i=k}^{l-1}h_{i}<\eta h_{k}$. Given a real $\varepsilon'>0$, choose $N$ with $\eta h_{N}<\varepsilon'$ (possible as $h_{k}\to0$); since $h_{k}\le h_{N}$ for $k\ge N$, $W_{2}(\gamma_{k},\gamma_{l})<\varepsilon'$ for all $k,l\ge N$ (for $k=l$ the distance is $0$). So $(\gamma_{k})$ is a Cauchy sequence in $(\Sigma_{2d},W_{2})$ (Cauchy Sequence in a Metric Space). By induction on $k$, $I(\gamma_{k})^{1/2}\le I(\gamma_{1})^{1/2}+\sum_{i=1}^{k-1}(a_{i}+b_{i})<w+\eta/4+\frac{\eta}{2}\sum_{i=1}^{k-1}h_{i}\le w+\eta$ (the sum being empty for $k=1$), hence

(f) The limit. Let $\gamma=[(\gamma_{k})_{k}]\in\Sigma^{2}_{2d}$; by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §density, $\kappa_{2d}(\gamma_{k})\to\gamma$. For $\epsilon=1,2$, $\mathrm{pr}^{\epsilon}_{\#}$ is continuous by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §lipschitz, so $\mathrm{pr}^{\epsilon}_{\#}\kappa_{2d}(\gamma_{k})\to\mathrm{pr}^{\epsilon}_{\#}\gamma$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential. By Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded and Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §coordinate, $\mathrm{pr}^{1}_{\#}\kappa_{2d}(\gamma_{k})=\kappa_{d}(\gamma_{k}\circ\sigma_{\mathrm{pr}^{1}})=\kappa_{d}(\gamma_{k}\circ\iota^{1})=\kappa_{d}(\mu_{k})$, which tends to $\mu$ by (a); so $\mathrm{pr}^{1}_{\#}\gamma=\mu$ by Uniqueness of Limits in a Metric Space, and in the same way $\mathrm{pr}^{2}_{\#}\gamma=\nu$. Thus $\gamma\in\Pi^{2}(\mu,\nu)$ by Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §couplings. Next, $D_{\#}$ and $\widehat{M}$ are continuous by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §lipschitz, so two applications of Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential give $\mathcal{I}(\kappa_{2d}(\gamma_{k}))=\widehat{M}(D_{\#}\kappa_{2d}(\gamma_{k}))\to\widehat{M}(D_{\#}\gamma)=\mathcal{I}(\gamma)$ (Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §cost). Since $\gamma_{k}\in\Pi(\mu_{k},\nu_{k})$, Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded gives $\mathcal{I}(\kappa_{2d}(\gamma_{k}))=I(\gamma_{k})\le w^{2}+\varepsilon$ for every $k$, and claim 1 of Order Properties of Limits of Real Sequences (against the constant sequence $w^{2}+\varepsilon$) gives $\mathcal{I}(\gamma)\le\widehat{W}_{2}(\mu,\nu)^{2}+\varepsilon$.

Proof of clause 2 (Infimum). By clause 1 with $\varepsilon=1$, $\Pi^{2}(\mu,\nu)$ is nonempty; let $E=\{\mathcal{I}(\gamma):\gamma\in\Pi^{2}(\mu,\nu)\}$. By Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §cost, $\widehat{W}_{2}(\mu,\nu)^{2}\le\mathcal{I}(\gamma)$ for every $\gamma\in\Pi^{2}(\mu,\nu)$, so $\widehat{W}_{2}(\mu,\nu)^{2}$ is a lower bound of $E$. Let $\ell$ be any lower bound of $E$ and suppose $\ell>\widehat{W}_{2}(\mu,\nu)^{2}$. Clause 1 with $\varepsilon=(\ell-\widehat{W}_{2}(\mu,\nu)^{2})/2>0$ gives $\gamma\in\Pi^{2}(\mu,\nu)$ with $\mathcal{I}(\gamma)\le\widehat{W}_{2}(\mu,\nu)^{2}+\varepsilon<\ell$, contradicting that $\ell$ is a lower bound. Since the order of $\mathbb{R}$ is total, $\ell\le\widehat{W}_{2}(\mu,\nu)^{2}$. By Lower Bound and Greatest Lower Bound in a Totally Ordered Set, $\widehat{W}_{2}(\mu,\nu)^{2}$ is the infimum of $E$.