Proof of The Second-Order Test-Datum Estimate at a Sequentially Strict Maximum of the Doubled Function

Each result cited is universally quantified over the data in its own statement. Write $\Theta:V\times V\to\mathbb{R}$ for the function in the hypothesis, so that $\Theta$ attains a sequentially strict maximum on $V\times V$ at $(\hat{x},\hat{y})$, and put $\bar{p}=\alpha(\hat{x}-\hat{y})\in H$. Local maxima and minima relative to $V$ are those of Real Hilbert Spaces: Standing Notation and Background §local-extrema for the subset $V$ of the metric space $(H,d_{H})$, which is the notion used both in Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space with $E=H$ and $A=V$ and in Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple with $U=H$, since $V\cap H=V$.

The order of choice. The data $\delta,\alpha,\sigma,\varepsilon,B,G,R$, the vectors $p,q$, the point $(\hat{x},\hat{y})$, the basis $(e_{k})_{k\in\mathbb{N}}$, the constant $\lambda$, the pair $(\omega_{1},\omega_{2})$ and the modulus $\omega$ are given by hypothesis. We fix next the two positive real numbers

$$\varepsilon_{1}=\tfrac{\varepsilon}{3},\qquad \rho=\tfrac{\varepsilon}{6},$$

and then construct, for every $m\in\mathbb{N}$ and in terms of the data already fixed alone, a family of test data indexed by $m$ (Steps 2 to 6). Only afterwards is the tail-insensitivity hypothesis applied to the resulting sequences, yielding an index $m$ at which the conclusion is read off (Steps 7 to 9). Nothing entering the construction of the $m$-th datum depends on that final index, so the choices are not circular.

Step 1 (two auxiliary functions). Let $\tilde{u},\tilde{v}:V\to\mathbb{R}$ be given by

$$\tilde{u}(x)=u^{-}_{\delta}(x)-\langle x,p\rangle_{H},\qquad \tilde{v}(y)=v^{+}_{\delta}(y)+\langle y,q\rangle_{H},$$

which agree with $u^{-}_{\delta}(x)-\langle p,x\rangle_{H}$ and $v^{+}_{\delta}(y)+\langle q,y\rangle_{H}$ by the symmetry of the inner product in Real Inner Product Space §inner-product, and let $-\tilde{v}:V\to\mathbb{R}$ have value $-\tilde{v}(y)$ at $y$. For all $x,y\in V$,

$$\tilde{u}(x)-\tilde{v}(y)-\tfrac{\alpha}{2}|x-y|_{H}^{2}=\Theta(x,y).$$

Bounds. By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity §bound, $u^{-}_{\delta}(x)\le C-\tfrac{\delta}{2}|x|_{H}^{2}$ and $-v^{+}_{\delta}(y)\le C-\tfrac{\delta}{2}|y|_{H}^{2}$ for $x,y\in V$. By The Cauchy-Schwarz Inequality in a Real Inner Product Space and $|p|_{H}\le\sigma\le1$ we have $-\langle x,p\rangle_{H}\le|x|_{H}|p|_{H}\le|x|_{H}$, and likewise $-\langle y,q\rangle_{H}\le|y|_{H}$. For every real $t$ the inequality $0\le\tfrac{\delta}{2}\bigl(t-\tfrac{1}{\delta}\bigr)^{2}$ expands to $t\le\tfrac{\delta}{2}t^{2}+\tfrac{1}{2\delta}$, so

$$\tilde{u}(x)\le C-\tfrac{\delta}{2}|x|_{H}^{2}+|x|_{H}\le C+\tfrac{1}{2\delta},\qquad -\tilde{v}(y)\le C-\tfrac{\delta}{2}|y|_{H}^{2}+|y|_{H}\le C+\tfrac{1}{2\delta}$$

for all $x,y\in V$. Hence the sets of values of $\tilde{u}$ and of $-\tilde{v}$ are bounded above.

Closed superlevel sets. By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity §closed-superlevel, applied with $U=H$, the functions $u^{-}_{\delta}$ and $-v^{+}_{\delta}$ on $V$ have closed superlevel sets in $(H,d_{H})$. The maps $x\mapsto\langle x,p\rangle_{H}$ and $y\mapsto\langle y,q\rangle_{H}$ from $H$ to $\mathbb{R}$ are Lipschitz by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §lipschitz, hence continuous on $H$ by A Lipschitz Map is Uniformly Continuous. Since $\tilde{u}$ is $u^{-}_{\delta}$ minus the first of these and $-\tilde{v}$ is $-v^{+}_{\delta}$ minus the second, Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits §perturbation shows that $\tilde{u}$ and $-\tilde{v}$ have closed superlevel sets in $(H,d_{H})$.

Step 2 (the doubling lemma at each truncation). Let $m\in\mathbb{N}$, let $e^{(m)}\in H^{m}$ be the $m$-tuple with components $e_{1},\dots,e_{m}$ and let $N_{m}\in\mathrm{Sym}(H)$ be its tail form, as in The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple §tail-forms; thus $e^{(m)}$ is orthonormal and all its components lie in $V$, which is a linear subspace of $H$ by Hilbert Triples: Standing Notation and Background §triple. By Step 1 the functions $\tilde{u}$ and $-\tilde{v}$ are bounded above and have closed superlevel sets in $H$, and $\Theta$ attains a sequentially strict maximum on $V\times V$ at $(\hat{x},\hat{y})$. Hence Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference applies with the space $H$, the subspace $V$, the tuple $e^{(m)}$, the functions $\tilde{u}$ and $\tilde{v}$ and the positive number $\alpha$, and provides $X_{m},Y_{m}\in\mathrm{Sym}(H)$ with the following properties.

By Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference §test-data, the quadruple $\bigl(\hat{x},\tilde{u}(\hat{x}),\bar{p},X_{m}+2\alpha N_{m}\bigr)$ is approximable by test data from above for $\tilde{u}$ on $V$, and the quadruple $\bigl(\hat{y},\tilde{v}(\hat{y}),\bar{p},Y_{m}-2\alpha N_{m}\bigr)$ is approximable by test data from below for $\tilde{v}$ on $V$.

By Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference §quadratic-bound, Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference §ordering and Lions' Lemma on a Hilbert Space: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference §norm-bound, the pair $(X_{m},Y_{m})$ satisfies $X_{m}\preceq Y_{m}$, $\lVert X_{m}\rVert\le6\alpha$, $\lVert Y_{m}\rVert\le6\alpha$ and $-3\alpha(|z|_{H}^{2}+|w|_{H}^{2})\le X_{m}(z,z)-Y_{m}(w,w)\le3\alpha|z-w|_{H}^{2}$ for all $z,w\in H$; that is, $(X_{m},Y_{m})$ is admitted at $\alpha$.

Finally $\lVert N_{m}\rVert\le1$ by Coordinate Maps of a Finite Orthonormal Tuple: Forms, the Tail Form, and Quadratic Test Functions §tail, so by the triangle inequality and the homogeneity of the norm in Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §norm-axioms,

$$\lVert X_{m}+2\alpha N_{m}\rVert\le\lVert X_{m}\rVert+2\alpha\lVert N_{m}\rVert\le6\alpha+2\alpha=8\alpha,$$

and in the same way $\lVert Y_{m}-2\alpha N_{m}\rVert\le8\alpha$.

Step 3 (test functions and test data at the index $m$). Let $\varepsilon_{m}$ be the least of the two positive real numbers $\varepsilon_{1}$ and $\tfrac{1}{m}$, which exists by claim 9 of Elementary Order Arithmetic in an Ordered Field; then $0<\varepsilon_{m}$, $\varepsilon_{m}\le\varepsilon_{1}=\tfrac{\varepsilon}{3}\le\tfrac{1}{3}$ and $\varepsilon_{m}\le\tfrac{1}{m}$.

By Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space §above, applied to the first quadruple of Step 2 with the tolerance $\varepsilon_{m}$, there are $z_{m}\in V$ and $\varphi_{m}\in C^{2}(H)$ such that the function on $V$ with value $\tilde{u}(x)-\varphi_{m}(x)$ at $x$ has a local maximum relative to $V$ at $z_{m}$ and

$$|z_{m}-\hat{x}|_{H}<\varepsilon_{m},\quad |\tilde{u}(z_{m})-\tilde{u}(\hat{x})|<\varepsilon_{m},\quad |D\varphi_{m}(z_{m})-\bar{p}|_{H}<\varepsilon_{m},\quad \lVert D^{2}\varphi_{m}(z_{m})-(X_{m}+2\alpha N_{m})\rVert<\varepsilon_{m}.$$

Let $\varphi'_{m}:H\to\mathbb{R}$ be given by $\varphi'_{m}(x)=\varphi_{m}(x)+\langle x,p\rangle_{H}$. The second summand belongs to $C^{2}(H)$ with gradient $p$ and Hessian $0_{\mathrm{Sym}}$ at every point of $H$ by Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §affine, so $\varphi'_{m}\in C^{2}(H)$ with $D\varphi'_{m}(z_{m})=D\varphi_{m}(z_{m})+p$ and $D^{2}\varphi'_{m}(z_{m})=D^{2}\varphi_{m}(z_{m})$ by Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum. Since $\tilde{u}(x)-\varphi_{m}(x)=u^{-}_{\delta}(x)-\varphi'_{m}(x)$ for every $x\in V$, the function on $V$ with value $u^{-}_{\delta}(x)-\varphi'_{m}(x)$ at $x$ has a local maximum relative to $V$ at $z_{m}$.

Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §subsolution with $\delta$, the test function $\varphi'_{m}$, the point $z_{m}$ and the number $\varepsilon_{m}$ yields $a_{m}\in D(A)$, $s_{m}\in\mathbb{R}$, $\pi_{m}\in H$ and $Z_{m}\in\mathrm{Sym}(H)$ with

$$|a_{m}-z_{m}|_{H}<\varepsilon_{m},\quad |u^{-}_{\delta}(a_{m})-u^{-}_{\delta}(z_{m})|<\varepsilon_{m},\quad |s_{m}-u^{-}_{\delta}(z_{m})|<\varepsilon_{m},$$ $$|\pi_{m}-D\varphi'_{m}(z_{m})|_{H}<\varepsilon_{m},\quad \lVert Z_{m}-D^{2}\varphi'_{m}(z_{m})\rVert<\varepsilon_{m},\quad F^{-}_{\delta}(a_{m},s_{m},\pi_{m},Z_{m})\le\varepsilon_{m}.$$

Symmetrically, Quadruple Approximable by Test Data on a Subset of a Real Inner Product Space §below, applied to the second quadruple of Step 2 with the tolerance $\varepsilon_{m}$, gives $w_{m}\in V$ and $\psi_{m}\in C^{2}(H)$ such that the function with value $\tilde{v}(y)-\psi_{m}(y)$ at $y$ has a local minimum relative to $V$ at $w_{m}$ and

$$|w_{m}-\hat{y}|_{H}<\varepsilon_{m},\quad |\tilde{v}(w_{m})-\tilde{v}(\hat{y})|<\varepsilon_{m},\quad |D\psi_{m}(w_{m})-\bar{p}|_{H}<\varepsilon_{m},\quad \lVert D^{2}\psi_{m}(w_{m})-(Y_{m}-2\alpha N_{m})\rVert<\varepsilon_{m}.$$

With $\psi'_{m}(y)=\psi_{m}(y)-\langle y,q\rangle_{H}$ we obtain, by the same two lemmas on differences (Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §affine and Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §difference), that $\psi'_{m}\in C^{2}(H)$ with $D\psi'_{m}(w_{m})=D\psi_{m}(w_{m})-q$ and $D^{2}\psi'_{m}(w_{m})=D^{2}\psi_{m}(w_{m})$; and $\tilde{v}(y)-\psi_{m}(y)=v^{+}_{\delta}(y)-\psi'_{m}(y)$ for $y\in V$, so the function with value $v^{+}_{\delta}(y)-\psi'_{m}(y)$ at $y$ has a local minimum relative to $V$ at $w_{m}$. Applying Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §supersolution with $\delta$, $\psi'_{m}$, $w_{m}$ and $\varepsilon_{m}$ yields $b_{m}\in D(A)$, $t_{m}\in\mathbb{R}$, $\kappa_{m}\in H$ and $W_{m}\in\mathrm{Sym}(H)$ with

$$|b_{m}-w_{m}|_{H}<\varepsilon_{m},\quad |v^{+}_{\delta}(b_{m})-v^{+}_{\delta}(w_{m})|<\varepsilon_{m},\quad |t_{m}-v^{+}_{\delta}(w_{m})|<\varepsilon_{m},$$ $$|\kappa_{m}-D\psi'_{m}(w_{m})|_{H}<\varepsilon_{m},\quad \lVert W_{m}-D^{2}\psi'_{m}(w_{m})\rVert<\varepsilon_{m},\quad -\varepsilon_{m}\le F^{+}_{\delta}(b_{m},t_{m},\kappa_{m},W_{m}).$$

Step 4 (elementary consequences at the index $m$). Fix $m\in\mathbb{N}$ and write $\varepsilon'=\varepsilon_{m}$. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle,

$$|a_{m}-\hat{x}|_{H}\le|a_{m}-z_{m}|_{H}+|z_{m}-\hat{x}|_{H}<2\varepsilon',\qquad |b_{m}-\hat{y}|_{H}<2\varepsilon' .$$

Since $u^{-}_{\delta}(z_{m})-u^{-}_{\delta}(\hat{x})=\bigl(\tilde{u}(z_{m})-\tilde{u}(\hat{x})\bigr)+\langle z_{m}-\hat{x},p\rangle_{H}$ by Elementary Identities in a Real Inner Product Space §bilinear, and $|\langle z_{m}-\hat{x},p\rangle_{H}|\le|z_{m}-\hat{x}|_{H}|p|_{H}\le\varepsilon'\sigma\le\varepsilon'$ by The Cauchy-Schwarz Inequality in a Real Inner Product Space, we get $|u^{-}_{\delta}(z_{m})-u^{-}_{\delta}(\hat{x})|<2\varepsilon'$ and therefore

$$|u^{-}_{\delta}(a_{m})-u^{-}_{\delta}(\hat{x})|<3\varepsilon',\qquad |s_{m}-u^{-}_{\delta}(\hat{x})|<3\varepsilon' ;$$

symmetrically $|v^{+}_{\delta}(b_{m})-v^{+}_{\delta}(\hat{y})|<3\varepsilon'$ and $|t_{m}-v^{+}_{\delta}(\hat{y})|<3\varepsilon'$. Since $D\varphi'_{m}(z_{m})-(\bar{p}+p)=D\varphi_{m}(z_{m})-\bar{p}$,

$$|\pi_{m}-(\bar{p}+p)|_{H}\le|\pi_{m}-D\varphi'_{m}(z_{m})|_{H}+|D\varphi_{m}(z_{m})-\bar{p}|_{H}<2\varepsilon' ,$$

and symmetrically $|\kappa_{m}-(\bar{p}-q)|_{H}<2\varepsilon'$. Finally $D^{2}\varphi'_{m}(z_{m})=D^{2}\varphi_{m}(z_{m})$ gives

$$\lVert Z_{m}-(X_{m}+2\alpha N_{m})\rVert<2\varepsilon',\qquad \lVert W_{m}-(Y_{m}-2\alpha N_{m})\rVert<2\varepsilon' .$$

As $3\varepsilon'\le3\varepsilon_{1}=\varepsilon$, the four quantities $|a_{m}-\hat{x}|_{H}$, $|b_{m}-\hat{y}|_{H}$, $|u^{-}_{\delta}(a_{m})-u^{-}_{\delta}(\hat{x})|$ and $|v^{+}_{\delta}(b_{m})-v^{+}_{\delta}(\hat{y})|$ are all less than $\varepsilon$.

Step 5 (the data are $R$-bounded, and two of them admissible). Keep $m$ fixed, write $\varepsilon'=\varepsilon_{m}$, and put $P_{m}=\alpha(a_{m}-b_{m})\in H$.

By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity §bound, $u^{-}_{\delta}(a_{m})\le C-\delta h(a_{m})$, so, using $C\le B$, $-u^{-}_{\delta}(\hat{x})\le B$ and $3\varepsilon'\le\varepsilon\le1$,

$$\delta h(a_{m})\le C-u^{-}_{\delta}(a_{m})\le C-u^{-}_{\delta}(\hat{x})+3\varepsilon'\le2B+1 ,$$

whence $h(a_{m})\le\tfrac{2B+1}{\delta}<R$. By the same claim $-C+\delta h(b_{m})\le v^{+}_{\delta}(b_{m})$, so $\delta h(b_{m})\le v^{+}_{\delta}(\hat{y})+3\varepsilon'+C\le2B+1$ and $h(b_{m})<R$. Next $|s_{m}|\le|u^{-}_{\delta}(\hat{x})|+3\varepsilon'\le B+1<R$ and $|t_{m}|\le B+1<R$, because $B+1\le3B+2<R$ and $0\le B$.

For the gradient arguments, $|\bar{p}|_{H}=\alpha|\hat{x}-\hat{y}|_{H}\le G$ by Elementary Identities in a Real Inner Product Space §homogeneity, so, using $\varepsilon'\le\tfrac{1}{3}$, $\sigma\le1$ and $1<\alpha$,

$$|\pi_{m}|_{H}\le|\bar{p}|_{H}+|p|_{H}+2\varepsilon'\le G+1+\tfrac{2}{3}\le G+2\alpha<R ,$$

and likewise $|\kappa_{m}|_{H}<R$. Also, by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle and Step 4,

$$|P_{m}|_{H}\le\alpha\bigl(|a_{m}-\hat{x}|_{H}+|\hat{x}-\hat{y}|_{H}+|\hat{y}-b_{m}|_{H}\bigr)\le G+4\alpha\varepsilon'\le G+\tfrac{4\alpha}{3}<G+2\alpha<R .$$

For the form arguments, Step 2 and Step 4 give $\lVert Z_{m}\rVert\le\lVert X_{m}+2\alpha N_{m}\rVert+2\varepsilon'\le8\alpha+1<8\alpha+2<R$ and likewise $\lVert W_{m}\rVert<R$, while $\lVert X_{m}+2\alpha N_{m}\rVert\le8\alpha<R$ and $\lVert Y_{m}-2\alpha N_{m}\rVert\le8\alpha<R$.

Therefore, by Test Data for a Second-Order Equation Operator on a Hilbert Triple and the Admissible Sets §bounded, each of the four test data

$$\xi_{m}=(a_{m},s_{m},\pi_{m},Z_{m}),\quad \eta_{m}=(b_{m},t_{m},\kappa_{m},W_{m}),\quad \zeta_{m}=(a_{m},s_{m},P_{m},X_{m}+2\alpha N_{m}),\quad \vartheta_{m}=(b_{m},t_{m},P_{m},Y_{m}-2\alpha N_{m})$$

is $R$-bounded. Moreover, by the last inequalities of Step 3,

$$F^{-}_{\delta}(\xi_{m})-F^{+}_{\delta}(\eta_{m})\le\varepsilon_{m}+\varepsilon_{m}\le2<8\alpha+2<R ,$$

so $\xi_{m}\in S^{-}_{\delta,R}$ and $\eta_{m}\in S^{+}_{\delta,R}$ by Test Data for a Second-Order Equation Operator on a Hilbert Triple and the Admissible Sets §admissible, each of the two data serving as the witness required for the other.

Step 6 (moving onto the forms of the doubling lemma). Put

$$\tau^{(m)}_{1}=|P_{m}-\pi_{m}|_{H}+\bigl\lVert (X_{m}+2\alpha N_{m})-Z_{m}\bigr\rVert ,\qquad \tau^{(m)}_{2}=|P_{m}-\kappa_{m}|_{H}+\bigl\lVert (Y_{m}-2\alpha N_{m})-W_{m}\bigr\rVert ,$$

which are nonnegative. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle and Step 4,

$$|P_{m}-\pi_{m}|_{H}\le\alpha\bigl|(a_{m}-b_{m})-(\hat{x}-\hat{y})\bigr|_{H}+\bigl|\bar{p}+p-\pi_{m}\bigr|_{H}+|p|_{H}\le4\alpha\varepsilon_{m}+2\varepsilon_{m}+\sigma ,$$

and the second summand of $\tau^{(m)}_{1}$ is at most $2\varepsilon_{m}$, so, using $\varepsilon_{m}\le\tfrac{\varepsilon}{3}$ and $1\le2\alpha$,

$$\tau^{(m)}_{1}\le(4\alpha+4)\varepsilon_{m}+\sigma\le\tfrac{4\alpha+4}{3}\,\varepsilon+\sigma\le(2\alpha+1)\varepsilon+\sigma ,$$

the last step because $4\alpha+4\le6\alpha+3$. The same bound holds for $\tau^{(m)}_{2}$, since $|\kappa_{m}-(\bar{p}-q)|_{H}<2\varepsilon_{m}$ and $|q|_{H}\le\sigma$.

Applying the first condition of The Shift-Continuity Condition on Admissible Test Data §modulus to $\xi_{m}\in S^{-}_{\delta,R}$ with the perturbations $P_{m}-\pi_{m}\in H$ and $(X_{m}+2\alpha N_{m})-Z_{m}\in\mathrm{Sym}(H)$, whose norms sum to $\tau^{(m)}_{1}$, gives

$$F^{-}_{\delta}(\zeta_{m})\ \le\ F^{-}_{\delta}(\xi_{m})+\omega\bigl(\tau^{(m)}_{1}\bigr)\ \le\ \varepsilon_{m}+\omega\bigl(\tau^{(m)}_{1}\bigr),$$

and applying the second condition to $\eta_{m}\in S^{+}_{\delta,R}$ with the perturbations $P_{m}-\kappa_{m}$ and $(Y_{m}-2\alpha N_{m})-W_{m}$ gives

$$-\varepsilon_{m}-\omega\bigl(\tau^{(m)}_{2}\bigr)\ \le\ F^{+}_{\delta}(\eta_{m})-\omega\bigl(\tau^{(m)}_{2}\bigr)\ \le\ F^{+}_{\delta}(\vartheta_{m}).$$

Step 7 (removing the tail). Each of the estimates

$$|a_{m}-\hat{x}|_{H}<2\varepsilon_{m},\quad |b_{m}-\hat{y}|_{H}<2\varepsilon_{m},\quad |s_{m}-u^{-}_{\delta}(\hat{x})|<3\varepsilon_{m},\quad |t_{m}-v^{+}_{\delta}(\hat{y})|<3\varepsilon_{m},\quad |P_{m}-\bar{p}|_{H}\le4\alpha\varepsilon_{m}$$

of Steps 4 and 6 is of the form $D_{m}\le\tfrac{K}{m}$ for a nonnegative real $K$ not depending on $m$, because $\varepsilon_{m}\le\tfrac{1}{m}$. Given a positive real $\epsilon$, Existence of a Sequence of Positive Real Numbers with Limit Zero, applied with the positive real $\tfrac{\epsilon}{K+1}$, provides $m_{\ast}\in\mathbb{N}$ with $\tfrac{1}{m_{\ast}}<\tfrac{\epsilon}{K+1}$, and then $\tfrac{K}{m}\le\tfrac{K+1}{m}\le\tfrac{K+1}{m_{\ast}}<\epsilon$ for every $m\in\mathbb{N}$ with $m_{\ast}\le m$. By Convergent Sequence in a Metric Space the sequence $(a_{m})_{m\in\mathbb{N}}$ therefore converges in $H$ to $\hat{x}$, the sequence $(b_{m})_{m\in\mathbb{N}}$ converges in $H$ to $\hat{y}$, the sequences $(s_{m})_{m\in\mathbb{N}}$ and $(t_{m})_{m\in\mathbb{N}}$ converge in $\mathbb{R}$ to $u^{-}_{\delta}(\hat{x})$ and to $v^{+}_{\delta}(\hat{y})$, and $(P_{m})_{m\in\mathbb{N}}$ converges in $H$ to $\bar{p}$. All the $a_{m}$ and $b_{m}$ lie in $D(A)=W$.

By Step 5 the test datum $\zeta_{m}$ is $R$-bounded for every $m\in\mathbb{N}$. Hence The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple §lower, applied with the positive numbers $\beta=2\alpha$, $R$ and $\rho$, with $\delta$, and with the sequences $(a_{m})$, $(s_{m})$, $(P_{m})$ and $(X_{m})$, provides $m_{1}\in\mathbb{N}$ such that every $m\in\mathbb{N}$ with $m_{1}\le m$ satisfies

$$F^{-}_{\delta}\bigl(a_{m},s_{m},P_{m},X_{m}\bigr)\ \le\ F^{-}_{\delta}(\zeta_{m})+\rho .$$

Likewise $\vartheta_{m}$ is $R$-bounded for every $m$, so The Tail-Insensitivity Condition for an Equation Operator on a Hilbert Triple §upper, applied with the same $\beta$, $R$, $\rho$ and $\delta$ and with the sequences $(b_{m})$, $(t_{m})$, $(P_{m})$ and $(Y_{m})$, provides $m_{2}\in\mathbb{N}$ such that every $m\in\mathbb{N}$ with $m_{2}\le m$ satisfies

$$F^{+}_{\delta}(\vartheta_{m})-\rho\ \le\ F^{+}_{\delta}\bigl(b_{m},t_{m},P_{m},Y_{m}\bigr).$$

Fix from now on an $m\in\mathbb{N}$ with $m_{1}\le m$ and $m_{2}\le m$, and set $x_{1}=a_{m}$, $y_{1}=b_{m}$, $\tau_{1}=\tau^{(m)}_{1}$ and $\tau_{2}=\tau^{(m)}_{2}$. Combining the last two displays with Step 6,

$$F^{-}_{\delta}\bigl(x_{1},s_{m},P_{m},X_{m}\bigr)\le\varepsilon_{m}+\omega(\tau_{1})+\rho,\qquad -\varepsilon_{m}-\omega(\tau_{2})-\rho\le F^{+}_{\delta}\bigl(y_{1},t_{m},P_{m},Y_{m}\bigr).$$

Clause 1. Since $P_{m}=\alpha(a_{m}-b_{m})$ by its definition in Step 5, and $x_{1}=a_{m}$, $y_{1}=b_{m}$, we have $P_{m}=\alpha(x_{1}-y_{1})$. Take $s=s_{m}$, $t=t_{m}$, $X=X_{m}$ and $Y=Y_{m}$. The pair $(X_{m},Y_{m})$ is admitted at $\alpha$ by Step 2, and $|s_{m}-u^{-}_{\delta}(\hat{x})|<3\varepsilon_{m}\le\varepsilon$ and $|t_{m}-v^{+}_{\delta}(\hat{y})|<3\varepsilon_{m}\le\varepsilon$ by Step 4 and $\varepsilon_{m}\le\varepsilon_{1}=\tfrac{\varepsilon}{3}$. Moreover $\varepsilon_{m}+\rho\le\tfrac{\varepsilon}{3}+\tfrac{\varepsilon}{6}<\varepsilon$, so the last display gives

$$F^{-}_{\delta}\bigl(x_{1},s_{m},\alpha(x_{1}-y_{1}),X_{m}\bigr)\le\varepsilon+\omega(\tau_{1}),\qquad -\varepsilon-\omega(\tau_{2})\le F^{+}_{\delta}\bigl(y_{1},t_{m},\alpha(x_{1}-y_{1}),Y_{m}\bigr),$$

which, once the bounds on $x_{1},y_{1},\tau_{1},\tau_{2}$ recorded at the end of the proof are in place, is clause 1.

Step 8 (the structure condition). We have $x_{1},y_{1}\in D(A)=W$, $|t_{m}|\le B+1\le3B+2$, $1<\alpha$, $0<\delta<1$ and $P_{m}=\alpha(x_{1}-y_{1})$, and the pair $(X_{m},Y_{m})$ is admitted at $\alpha$ by Step 2. Hence The Second-Order Structure Condition for an Equation Operator on a Hilbert Triple §pair, applied to the second-order structure pair $(\omega_{1},\omega_{2})$ at $3B+2$ with the value argument $t_{m}$ and the form arguments $X_{m}$ and $Y_{m}$, gives

$$F^{+}_{\delta}\bigl(y_{1},t_{m},P_{m},Y_{m}\bigr)-\Omega\ \le\ F^{-}_{\delta}\bigl(x_{1},t_{m},P_{m},X_{m}\bigr),$$

where

$$\Omega=\omega_{1}\Bigl(\alpha|x_{1}-y_{1}|_{H}^{2}+\tfrac{1}{\alpha}\Bigr)+\omega_{2}\bigl(\delta\,(h(x_{1})+h(y_{1})+1),\,\alpha\bigr),$$

a nonnegative number because $\omega_{1}$ and the function $t\mapsto\omega_{2}(t,\alpha)$ are moduli of continuity, whose values are nonnegative by clause 1 of Modulus of Continuity.

Step 9 (properness and conclusion). Since $|s_{m}-u^{-}_{\delta}(\hat{x})|<3\varepsilon_{m}\le\varepsilon$ and $|t_{m}-v^{+}_{\delta}(\hat{y})|<3\varepsilon_{m}\le\varepsilon$,

$$u^{-}_{\delta}(\hat{x})-v^{+}_{\delta}(\hat{y})\le(s_{m}+\varepsilon)-(t_{m}-\varepsilon)=s_{m}-t_{m}+2\varepsilon . \tag{$*$}$$

Suppose first that $t_{m}\le s_{m}$. By Step 5 we have $0\le\delta h(x_{1})\le2B+1$, $|s_{m}|\le B+1$ and $|t_{m}|\le B+1$, so both $s_{m}+\delta h(x_{1})$ and $t_{m}+\delta h(x_{1})$ lie between $-(3B+2)$ and $3B+2$, and $t_{m}+\delta h(x_{1})\le s_{m}+\delta h(x_{1})$. Applying Locally Strictly Proper Second-Order Equation Operator on a Hilbert Triple §constant at the level $3B+2$, with the point $x_{1}$, the gradient argument $P_{m}+\delta Ax_{1}\in H$ and the form argument $X_{m}|_{V}+\delta I_{V}\in\mathrm{Sym}(V)$, and reading the two resulting values of $F$ through Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts §shifted, we obtain

$$\lambda(s_{m}-t_{m})=\lambda\bigl((s_{m}+\delta h(x_{1}))-(t_{m}+\delta h(x_{1}))\bigr)\le F^{-}_{\delta}\bigl(x_{1},s_{m},P_{m},X_{m}\bigr)-F^{-}_{\delta}\bigl(x_{1},t_{m},P_{m},X_{m}\bigr).$$

Combining this with Step 8 and the two inequalities at the end of Step 7,

$$\lambda(s_{m}-t_{m})\le\varepsilon_{m}+\omega(\tau_{1})+\rho-F^{+}_{\delta}\bigl(y_{1},t_{m},P_{m},Y_{m}\bigr)+\Omega\ \le\ 2\varepsilon_{m}+2\rho+\omega(\tau_{1})+\omega(\tau_{2})+\Omega .$$

Since $\varepsilon_{m}\le\varepsilon_{1}=\tfrac{\varepsilon}{3}$ and $\rho=\tfrac{\varepsilon}{6}$, we have $2\varepsilon_{m}+2\rho\le\tfrac{2\varepsilon}{3}+\tfrac{\varepsilon}{3}=\varepsilon$. Multiplying $(*)$ by the positive number $\lambda$ and adding $2\lambda\varepsilon$ to the previous display gives

$$\lambda\bigl(u^{-}_{\delta}(\hat{x})-v^{+}_{\delta}(\hat{y})\bigr)\le2\lambda\varepsilon+2\varepsilon+\omega(\tau_{1})+\omega(\tau_{2})+\Omega ,$$

which is the asserted estimate.

Suppose now that $s_{m}<t_{m}$. Then $(*)$ gives $u^{-}_{\delta}(\hat{x})-v^{+}_{\delta}(\hat{y})<2\varepsilon$, so $\lambda\bigl(u^{-}_{\delta}(\hat{x})-v^{+}_{\delta}(\hat{y})\bigr)<2\lambda\varepsilon$, and the asserted estimate holds because its remaining right-hand terms are nonnegative: the values of the modulus of continuity $\omega$ are nonnegative by clause 1 of Modulus of Continuity, $\Omega$ is nonnegative by Step 8, and $0<2\varepsilon$.

In both cases $x_{1},y_{1}\in D(A)$ satisfy $|x_{1}-\hat{x}|_{H}<\varepsilon$, $|y_{1}-\hat{y}|_{H}<\varepsilon$, $|u^{-}_{\delta}(x_{1})-u^{-}_{\delta}(\hat{x})|<\varepsilon$ and $|v^{+}_{\delta}(y_{1})-v^{+}_{\delta}(\hat{y})|<\varepsilon$ by Step 4, and $\tau_{1},\tau_{2}$ are nonnegative and at most $(2\alpha+1)\varepsilon+\sigma$ by Step 6. For these $x_{1},y_{1},\tau_{1},\tau_{2}$, clause 1 holds by the end of Step 7 and clause 2, the asserted estimate, holds by the two cases above; so both clauses are proved.