Proof of The Mean-Field Limit of the Relative Score of Laws of Dyson Particles Concentrating at a Measure

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here.

Conventions. For $k\in\mathbb{N}$ we write $N=N_{k}$ when $k$ is fixed, $\Sigma_{k}=\Sigma_{N_{k}}(P_{k})=\nabla P_{N}+a_{N}\xi_{P_{k}}$, $\langle\cdot,\cdot\rangle_{k}$ and $\lVert\cdot\rVert_{k}$ for the inner product and norm of $L^{2}(P_{k};\mathbb{R}^{N})$, and $\delta_{k}=\int_{\mathbb{R}^{N}}W_{2}(\mu^{N}_{x},\hat{\mu})^{2}\,P_{k}(dx)$, so that $\delta_{k}\to0$. As $(N_{k})$ is strictly increasing, $N_{k}\ge k$, so $N_{k}\to\infty$, $a_{N_{k}}\to0$ and $1/(N_{k}-1)\to0$. Since $P_{k}\in\mathcal{D}^{\Sigma}_{N}$, clauses The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set §energy and The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set §score give $P_{k}(W_{N})=1$, $P_{k}\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{N})$ with score $\xi_{P_{k}}$, and $\int\lVert\nabla P_{N}\rVert^{2}\,dP_{k}<\infty$. Also $A\ge0$, being an upper bound of a nonnegative number. By The Confined Logarithmic-Energy Pair on the Wasserstein Space of the Real Line §pair and The Logarithmic Energy of a Probability Measure on the Real Line §energy, $\hat{\mu}\in\mathcal{D}\subseteq\mathcal{D}_{\log}\subseteq\mathcal{P}_{2}(\mathbb{R})$. By The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §hypotheses, $V$ is of class $C^{2}$ with $V''\ge0$ and has regular growth; $V'$ is continuous and Borel by A Confining Potential and Its Derivative are Continuous and Borel §continuous and A Confining Potential and Its Derivative are Continuous and Borel §borel, and $V''$ is continuous by Confining Potentials on the Real Line §confining. Inner products in $L^{2}(P_{k};\mathbb{R}^{N})$ and $L^{2}(\hat{\mu};\mathbb{R})$ are integrals of dot products by Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu and One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars; these are real Hilbert spaces, and "Cauchy-Schwarz" refers to The Cauchy-Schwarz Inequality in a Real Inner Product Space in them. The letter $g$ of The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations is not used below.

Call $\varphi:\mathbb{R}\to\mathbb{R}$ admissible if it is differentiable at every point with continuous derivative $\varphi'$ and both $\varphi$ and $\varphi'$ are bounded; let $L_{\varphi}$ be a bound for $|\varphi'|$, let $\varphi^{\oplus}(x)=(\varphi(x_{1}),\dots,\varphi(x_{N}))$ (Borel and bounded, hence in $L^{2}(P_{k};\mathbb{R}^{N})$), and let $Q_{\varphi}:\mathbb{R}^{2}\to\mathbb{R}$ be $Q_{\varphi}(s,t)=\frac{\varphi(s)-\varphi(t)}{s-t}$ for $s\ne t$ and $Q_{\varphi}(s,s)=\varphi'(s)$. By The Difference Quotient of a Function with Bounded Continuous Derivative is Bounded, Symmetric and Continuous on the Plane §bound, The Difference Quotient of a Function with Bounded Continuous Derivative is Bounded, Symmetric and Continuous on the Plane §continuous and The Difference Quotient of a Function with Bounded Continuous Derivative is Bounded, Symmetric and Continuous on the Plane §measurable, $Q_{\varphi}$ is symmetric, continuous, Borel and bounded by $L_{\varphi}$. For $\psi\in C_{c}^{\infty}(\mathbb{R})$, $\psi'$ is admissible and $Q_{\psi'}=F_{\psi}$, by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives ($(\psi')'=\Delta\psi$ is continuous and bounded) and the definition of $F_{\psi}$ there.

Step 0 (Averages against empirical measures). Let $h:\mathbb{R}\to\mathbb{R}$ and $G:\mathbb{R}^{2}\to\mathbb{R}$ be bounded and Lipschitz with constant $L$ (Euclidean distance on $\mathbb{R}^{2}$). We claim: (a) for $x\in\mathbb{R}^{N}$,

the first identity holding for every Borel $h$ and the second for every bounded Borel $G$; and (b)

The first identity of (a) is Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §integral (with $d=1$ the $i$-th particle of $x$ is $x_{i}$). For bounded Borel $G$ and $\mu,\nu\in\mathcal{P}(\mathbb{R})$, applying Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product and the Tonelli part of Tonelli and Fubini Theorems to the positive and negative parts of $G$ gives, with all integrals finite,

with $\mu=\nu=\mu^{N}_{x}$ and the first identity twice this gives the second. For (b), let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R})$; Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment gives $\pi\in\Pi(\mu,\nu)$ with $I(\pi)=W_{2}(\mu,\nu)^{2}$, and Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §lipschitz gives $|\int h\,d\mu-\int h\,d\nu|\le L\,W_{2}(\mu,\nu)$. For fixed $s$ the function $t\mapsto G(s,t)$ is bounded and Lipschitz with constant $L$, and likewise $s\mapsto G(s,t)$ for fixed $t$; hence by $(*)$, integrating first in $t$ and then first in $s$,

Apply these with $\mu=\mu^{N}_{x}$, which lies in $\mathcal{P}_{2}(\mathbb{R})$ by Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §moment, and $\nu=\hat{\mu}$, and integrate against $P_{k}$: the functions of $x$ involved are Borel by (a) and Basic Properties of Empirical Measures: Values, Integrals, Push-Forwards, Second Moment, and Lipschitz Dependence on the Configuration §distance. Finally, for every positive $t$, $W\le\frac{1}{2}(t+W^{2}/t)$ for $W\ge0$, so $\int W_{2}(\mu^{N}_{x},\hat{\mu})\,P_{k}(dx)\le\frac{1}{2}(t+\delta_{k}/t)$; taking $t=\delta_{k}^{1/2}$ if $\delta_{k}>0$ and letting $t\to0$ otherwise gives the bound $\delta_{k}^{1/2}$, which proves (b).

(c) For $\psi\in C_{c}^{\infty}(\mathbb{R})$ the functions $(\psi')^{2}$ and $V'\psi'$ on $\mathbb{R}$ and $F_{\psi}$ on $\mathbb{R}^{2}$ are bounded and Lipschitz. Indeed, $\psi$ being smooth with compact support, $\psi',\psi'',\psi'''$ are continuous and vanish outside a compact set $K$, hence are bounded, say $|\psi'''|\le L_{3}$. $(\psi')^{2}$ and $V'\psi'$ are differentiable with derivatives $2\psi'\psi''$ and $V''\psi'+V'\psi''$ by Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives; these and the functions themselves are continuous and vanish off $K$, hence are bounded, and bounded derivatives give Lipschitz bounds by Mean Value Theorem on a Closed Real Interval. For $s\ne t$ the function $r\mapsto\psi'(t+r(s-t))$ is an antiderivative of $r\mapsto(s-t)\psi''(t+r(s-t))$ by Chain Rule for One-Dimensional Derivatives, the latter being continuous and hence Riemann integrable on $[0,1]$, so Fundamental Theorem of Calculus, Part II, on a Closed Real Interval on $[0,1]$ gives

which also holds for $s=t$, the integrand then being the constant $\Delta\psi(t)=\psi''(t)$. As $|\psi''(u)-\psi''(u')|\le L_{3}|u-u'|$ by Mean Value Theorem on a Closed Real Interval and $|(t+r(s-t))-(t'+r(s'-t'))|\le(1-r)|t-t'|+r|s-s'|\le\lVert(s,t)-(s',t')\rVert$ for $r\in[0,1]$, $F_{\psi}$ is Lipschitz with constant $L_{3}$; it is bounded by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §quotient.

Step 1 (Particle pairing identity). Let $\varphi$ be admissible and $k\in\mathbb{N}$, $N=N_{k}$. We claim that

is $P_{k}$-integrable and

By The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §hypotheses, $P_{N}$ is the function $P$ of The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality with strength $b_{N}$ and $V_{1}=V$; comparing The Confined Logarithmic Energy on the Weyl Chamber: Regularity, Penalty Property, Monotone Gradient and the Dissipation Inequality §regularity with The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §derivatives (strength $b_{N}$) gives $\partial_{i}P_{N}(x)=V'(x_{i})+\partial_{i}H_{b_{N}}(x)$ for $x\in W_{N}$. Hence The Logarithmic Energy on the Weyl Chamber: Derivatives, Monotone Gradient, the Calogero Identity and the Euler Identity §symmetrisation, with strength $b_{N}$ and $z=\varphi^{\oplus}(x)$, gives for $x\in W_{N}$, since $a_{ii}=0$ and $a_{ij}(x)(\varphi(x_{i})-\varphi(x_{j}))=Q_{\varphi}(x_{i},x_{j})$ for $i\ne j$,

The left side is $P_{k}$-integrable ($\nabla P_{N},\varphi^{\oplus}\in L^{2}(P_{k};\mathbb{R}^{N})$), $P_{k}(W_{N})=1$, $|Q_{\varphi}|\le L_{\varphi}$ and $V'$ is Borel; hence $x\mapsto\sum_{i}V'(x_{i})\varphi(x_{i})$ is $P_{k}$-integrable and $\langle\nabla P_{N},\varphi^{\oplus}\rangle_{k}=\int\bigl[\sum_{i}V'(x_{i})\varphi(x_{i})-\frac{b_{N}}{2}\sum_{i\ne j}Q_{\varphi}(x_{i},x_{j})\bigr]P_{k}(dx)$.

For the score, let $\chi_{R}$ ($R>0$) and $M_{1}$ be the cutoffs and the constant of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff with $q=N$. The field $\chi_{R}\varphi^{\oplus}$, with components $x\mapsto\chi_{R}(x)\varphi(x_{i})$, is of class $C^{1}$ on $\mathbb{R}^{N}$ and compactly supported, with divergence $\sum_{i}\partial_{i}\chi_{R}(x)\varphi(x_{i})+\chi_{R}(x)\sum_{i}\varphi'(x_{i})$. By The Score Integrates by Parts Against Every Compactly Supported Continuously Differentiable Vector Field §parts, applied at the configuration level to $P_{k}$,

Take $R=m\in\mathbb{N}$ and let $m\to\infty$: $\chi_{m}(x)=1$ once $m\ge\lVert x\rVert$, $|\chi_{m}\xi_{P_{k}}\cdot\varphi^{\oplus}|\le|\xi_{P_{k}}\cdot\varphi^{\oplus}|$, which is integrable, $|\sum_{i}\partial_{i}\chi_{m}(x)\varphi(x_{i})|\le NM_{1}\sup|\varphi|/m$ and $|\chi_{m}\sum_{i}\varphi'(x_{i})|\le NL_{\varphi}$. By Dominated Convergence Theorem, $\langle\xi_{P_{k}},\varphi^{\oplus}\rangle_{k}=-\int\sum_{i}\varphi'(x_{i})\,P_{k}(dx)$. Adding $a_{N}$ times this to the previous display gives (E1).

Normalised form. By Step 0(a), $\sum_{i\ne j}Q_{\varphi}(x_{i},x_{j})=N^{2}\int Q_{\varphi}\,d(\mu^{N}_{x}\boxtimes\mu^{N}_{x})-\sum_{i}\varphi'(x_{i})$ and $\sum_{i}V'(x_{i})\varphi(x_{i})=N\int V'\varphi\,d\mu^{N}_{x}$; since $\frac{b_{N}}{2N}N^{2}=\frac{\beta N}{4(N-1)}$, (E1) becomes

with $e_{k}(\varphi)=\bigl(\frac{\beta}{4N(N-1)}-\frac{a_{N}}{N}\bigr)\int\sum_{i}\varphi'(x_{i})\,P_{k}(dx)$, so that $|e_{k}(\varphi)|\le L_{\varphi}\bigl(\frac{\beta}{4(N-1)}+a_{N}\bigr)\to0$ as $k\to\infty$.

Step 2 (Limits along test functions). For $\psi\in C_{c}^{\infty}(\mathbb{R})$ let $\ell(\psi)=\int V'\psi'\,d\hat{\mu}-\frac{\beta}{4}\int F_{\psi}\,d(\hat{\mu}\boxtimes\hat{\mu})$ (both integrands bounded Borel). We claim

Apply (E2) to $\varphi=\psi'$, for which $Q_{\psi'}=F_{\psi}$. By Step 0(b),(c), $\int\int V'\psi'\,d\mu^{N}_{x}\,P_{k}(dx)\to\int V'\psi'\,d\hat{\mu}$ and $c_{k}=\int\int F_{\psi}\,d(\mu^{N}_{x}\boxtimes\mu^{N}_{x})\,P_{k}(dx)\to\int F_{\psi}\,d(\hat{\mu}\boxtimes\hat{\mu})$; as $(c_{k})$ is bounded (One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §quotient) and $\frac{N}{N-1}\to1$, and $e_{k}(\psi')\to0$, the first limit follows. Next $\lVert\psi'^{\oplus}(x)\rVert^{2}=\sum_{i}\psi'(x_{i})^{2}=N\int(\psi')^{2}\,d\mu^{N}_{x}$, so the second limit is Step 0(b),(c) together with $\lVert\psi'\rVert_{\hat{\mu}}^{2}=\int(\psi')^{2}\,d\hat{\mu}$ (One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives). Finally, by Cauchy-Schwarz and the hypothesis,

and letting $k\to\infty$ gives the third claim.

Step 3 ($V'$ is square-integrable under $\hat{\mu}$). By regular growth (The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §hypotheses) with $\eta=1/\beta$ there is $C'$ with $V''\le C'+V'^{2}/\beta$; let $C=\max(C',0)$. For $m\in\mathbb{N}$ let $\Theta_{m}(s)=s\,(1+s^{2}/m^{2})^{-1/2}$ and $\varphi_{m}=\Theta_{m}\circ V'$. Elementary calculus gives: $\Theta_{m}$ is differentiable with continuous derivative $\Theta_{m}'(s)=(1+s^{2}/m^{2})^{-3/2}\in(0,1]$; $\Theta_{m}(s)^{2}=\frac{m^{2}s^{2}}{m^{2}+s^{2}}\le m^{2}$; $\Theta_{m}(s)^{2}=\frac{s^{2}}{1+s^{2}/m^{2}}\le\frac{s^{2}}{(1+s^{2}/m^{2})^{1/2}}=s\,\Theta_{m}(s)$; and $\Theta_{m}(s)^{2}$ is nondecreasing in $m$ with limit $s^{2}$. By Chain Rule for One-Dimensional Derivatives, $\varphi_{m}$ is differentiable with continuous derivative $\varphi_{m}'=\Theta_{m}'(V')\,V''$, and by $V''\ge0$ and $(1+u)^{-3/2}\le(1+u)^{-1}$ for $u\ge0$,

Thus $\varphi_{m}$ is admissible with $L_{\varphi_{m}}=C+m^{2}/\beta$, and nondecreasing by Mean Value Theorem on a Closed Real Interval. For $s\ne t$ the same theorem gives $r$ between $s$ and $t$ with $Q_{\varphi_{m}}(s,t)=\varphi_{m}'(r)$, and $\varphi_{m}(r)$ lies between $\varphi_{m}(s)$ and $\varphi_{m}(t)$, so $\varphi_{m}(r)^{2}\le\varphi_{m}(s)^{2}+\varphi_{m}(t)^{2}$; with (E3), also on the diagonal,

Fix $k$, $N=N_{k}$, and let $X_{k}=\frac{1}{N}\int\sum_{i}\varphi_{m}(x_{i})^{2}\,P_{k}(dx)=\frac{1}{N}\lVert\varphi_{m}^{\oplus}\rVert_{k}^{2}$. Since $V'\varphi_{m}=V'\Theta_{m}(V')\ge\varphi_{m}^{2}$, (E1), (E4) and (E3) give

As $\frac{b_{N}(N-1)}{2}=\frac{\beta}{4}$, dividing by $N$ and using Cauchy-Schwarz as in Step 2, $\frac{1}{N}\langle\Sigma_{k},\varphi_{m}^{\oplus}\rangle_{k}\le(AX_{k})^{1/2}\le A+\frac{X_{k}}{4}$, we get $X_{k}\le A+\frac{X_{k}}{4}+\frac{\beta C}{4}+\frac{X_{k}}{2}+a_{N}(C+\frac{m^{2}}{\beta})$, that is,

As $a_{N_{k}}\to0$, $X_{k}\le C^{*}:=4A+\beta C+1$ for all large $k$, with $C^{*}$ independent of $m$. The function $\varphi_{m}^{2}$ is bounded by $m^{2}$ and Lipschitz (its derivative $2\varphi_{m}\varphi_{m}'$ is bounded by $2m(C+m^{2}/\beta)$; Mean Value Theorem on a Closed Real Interval), and $X_{k}=\int\int\varphi_{m}^{2}\,d\mu^{N}_{x}\,P_{k}(dx)$ by Step 0(a); so Step 0(b) gives $\int\varphi_{m}^{2}\,d\hat{\mu}=\lim_{k}X_{k}\le C^{*}$ for every $m$. Since $\varphi_{m}^{2}=\Theta_{m}(V')^{2}$ increases in $m$ to $V'^{2}$ pointwise, Monotone Convergence Theorem gives $\int V'^{2}\,d\hat{\mu}\le C^{*}<\infty$.

Step 4 (Clause 1, the limit has a score). For $\psi\in C_{c}^{\infty}(\mathbb{R})$, by the definition of $\ell$, Cauchy-Schwarz in $L^{2}(\hat{\mu};\mathbb{R})$ (where $V'$ lies by Step 3) and Step 2,

and $\lVert\psi'\rVert_{\hat{\mu}}=\lVert\nabla\psi\rVert_{\hat{\mu}}$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives. So $\hat{\mu}\in\mathcal{P}_{2}(\mathbb{R})$ has finite free Fisher information, with constant $\frac{4}{\beta}(\lVert V'\rVert_{\hat{\mu}}+A^{1/2})$; with $\hat{\mu}\in\mathcal{D}$ and Step 3, $\hat{\mu}\in\mathcal{D}_{\Sigma}$ by The Confined Logarithmic-Energy Pair on the Wasserstein Space of the Real Line §pair. By that clause, Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line §score and One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives, for every $\psi\in C_{c}^{\infty}(\mathbb{R})$

By The Confined Logarithmic-Energy Pair on the Wasserstein Space of the Real Line §pair, $\Sigma(\hat{\mu})\in T_{\hat{\mu}}$, which is the closure of the set of classes of the $\nabla\psi=\psi'$ by The Tangent Space of the Wasserstein Space at a Probability Measure §gradients and The Tangent Space of the Wasserstein Space at a Probability Measure §tangent. Choose $\psi_{n}\in C_{c}^{\infty}(\mathbb{R})$ with $\lVert\Sigma(\hat{\mu})-\psi_{n}'\rVert_{\hat{\mu}}\to0$; then $\langle\Sigma(\hat{\mu}),\psi_{n}'\rangle_{\hat{\mu}}\to\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}^{2}$ and $\lVert\psi_{n}'\rVert_{\hat{\mu}}\to\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}$, while by (E5) and Step 2 $\langle\Sigma(\hat{\mu}),\psi_{n}'\rangle_{\hat{\mu}}=\ell(\psi_{n})\le A^{1/2}\lVert\psi_{n}'\rVert_{\hat{\mu}}$. Hence $\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}^{2}\le A^{1/2}\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}$, so $\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}\le A^{1/2}$ and $\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}^{2}\le A$.

Step 5 (Clause 2, convergence of pairings). Write $S=\Sigma(\hat{\mu})$. Given positive $\varepsilon$, take $\psi$ and $k_{0}$ as in the hypothesis. For $k\ge k_{0}$, by bilinearity, (E5), Cauchy-Schwarz, the bound on $\frac{1}{N}\lVert\Sigma_{k}\rVert_{k}^{2}$ and Step 4,

The last term tends to $0$ by Step 2, so the upper limit of the left side is at most $2A^{1/2}\varepsilon$; $\varepsilon$ being arbitrary, the first sequence converges to $\langle S,q\rangle_{\hat{\mu}}$. For the norms let $n_{k}=(\frac{1}{N}\lVert G_{k}\rVert_{k}^{2})^{1/2}$. For $k\ge k_{0}$ the triangle inequality in $L^{2}(P_{k};\mathbb{R}^{N})$ gives $|n_{k}-(\frac{1}{N}\lVert\psi'^{\oplus}\rVert_{k}^{2})^{1/2}|\le(\frac{1}{N}\lVert G_{k}-\psi'^{\oplus}\rVert_{k}^{2})^{1/2}<\varepsilon$, that in $L^{2}(\hat{\mu};\mathbb{R})$ gives $|\lVert\psi'\rVert_{\hat{\mu}}-\lVert q\rVert_{\hat{\mu}}|<\varepsilon$, and $(\frac{1}{N}\lVert\psi'^{\oplus}\rVert_{k}^{2})^{1/2}\to\lVert\psi'\rVert_{\hat{\mu}}$ by Step 2. So the upper limit of $|n_{k}-\lVert q\rVert_{\hat{\mu}}|$ is at most $2\varepsilon$ for every $\varepsilon$; hence $n_{k}\to\lVert q\rVert_{\hat{\mu}}$ and $n_{k}^{2}\to\lVert q\rVert_{\hat{\mu}}^{2}$.

Step 6 (Clause 3, lower limit of the Fisher term). Given positive $\varepsilon$, choose as in Step 4 some $\psi\in C_{c}^{\infty}(\mathbb{R})$ with $\lVert S-\psi'\rVert_{\hat{\mu}}^{2}<\varepsilon/2$. For every $k$, expanding $0\le\lVert\Sigma_{k}-\psi'^{\oplus}\rVert_{k}^{2}$,

By Step 2 and (E5), $r_{k}\to2\langle S,\psi'\rangle_{\hat{\mu}}-\lVert\psi'\rVert_{\hat{\mu}}^{2}=\lVert S\rVert_{\hat{\mu}}^{2}-\lVert S-\psi'\rVert_{\hat{\mu}}^{2}>\lVert S\rVert_{\hat{\mu}}^{2}-\varepsilon/2$. Hence there is $k_{0}$ with $r_{k}>\lVert S\rVert_{\hat{\mu}}^{2}-\varepsilon$ for $k\ge k_{0}$, and then $\lVert S\rVert_{\hat{\mu}}^{2}\le\frac{1}{N_{k}}\lVert\Sigma_{N_{k}}(P_{k})\rVert_{P_{k}}^{2}+\varepsilon$.