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At $\alpha =0$, bounded Hessians have no uniform modulus of continuity, so the Arzela-Ascoli theorem does not supply the compact $C^2$ limit used in the contradiction. At $\alpha =1$, the blow-up has cubic growth, and the Liouville theorem no longer forces its Hessian to be constant: nonzero harmonic cubic polynomials are possible. These are the two endpoint failures behind the restriction $0<\alpha <1$; ordinary Schauder theory replaces neither endpoint by a general $C^{2,\alpha }$ estimate of the displayed form.

SetPart (b.iii), applied on each compactly contained ball, gives uniform $C^{2,\alpha }$ bounds for $v_k$. A diagonal use of the Arzela-Ascoli theorem therefore gives a subsequence converging in $C^2(K)$ on every compact $K⋐B_1$ to a $C^2$ function $w$.

The same estimate applied to $u_k$ gives $D{u}_k\to 0$ locally uniformly because $\Vert u_k{\Vert }_2\to 0$. Dividing the minimal-surface equation by $\Vert u_k{\Vert }_2$ shows thatThe coefficients converge locally uniformly to ${\delta }_{ij}$. Passing to the $C^2$ limit yieldsThus $w$ is harmonic and the required normalized subsequence converges to it locally in $C^2$.

Choose $0<\alpha <\epsilon /p$. At dyadic level $m$, Markov inequality and a union bound giveuniformly in $n$. Summing over $m$ shows that, outside a set of probability at most $C_0{C}^{-p}$, all dyadic increments obey this bound. Chaining dyadic approximations and using continuity givesfor all $s,t$. Since $X_0^n=0$, the paths then lie in the stated compact Hölder set by the Arzela-Ascoli theorem. Taking $C$ large proves tightness.

The Arzela-Ascoli theorem in sequential form states: if $K$ is a compact metric space and $(f_n)$ is a uniformly bounded, equicontinuous sequence in $C(K)$, then it has a uniformly convergent subsequence. Equivalently, a subset of $C(K)$ is relatively compact in the uniform norm exactly when it is uniformly bounded and equicontinuous.

For sufficiency, choose a finite $1/m$-net $E_m\subset K$ for every $m$. The union $D={\bigcup }_m{E}_m$ is countable and dense. Uniform boundedness makes each scalar sequence $(f_n(x){)}_n$, $x\in D$, bounded. Successive applications of Bolzano-Weierstrass followed by a diagonal choice give a subsequence $(g_n)$ for which $g_n(x)$ converges at every $x\in D$.

Given $\epsilon >0$, equicontinuity supplies $\delta >0$ such thatfor every $n$. Choose a finite $\delta$-net from $D$. Pointwise convergence on those finitely many points makes $(g_n)$ uniformly Cauchy on the net, and the two equicontinuity estimates make it uniformly Cauchy on all of $K$. Since $C(K)$ is complete, $g_n$ converges uniformly.

Conversely, a relatively compact family is uniformly bounded because its closure is compact in the normed space $C(K)$. Given $\epsilon >0$, cover that compact closure by finitely many uniform balls of radius $\epsilon /3$, centred at continuous functions $h_1,\dots ,h_r$. Uniform continuity of the finitely many $h_i$ gives one $\delta$ that works for all of them. Approximating any family member by one $h_i$ proves equicontinuity.

Let the common bound be $M$. The derivative bound and the mean-value theorem giveso on every compact interval the sequence is uniformly bounded and equicontinuous. Apply the Arzela-Ascoli theorem first on $[-1,1]$, then to a subsequence on $[-2,2]$, and so on. The diagonal subsequence for locally uniform convergence gives a strictly increasing $\varphi$ such that $f_{\varphi (n)}$ converges uniformly on every $[-R,R]$ to a function $f$. The limit is continuous and satisfies $|f(x)|\le M$.

Global uniform convergence need not follow. Choose a nonzero continuously differentiable bump function $\psi$ supported in $[-1,1]$ and setThe functions and their derivatives have a common bound. On every fixed compact interval, $f_n$ is eventually zero, so every subsequence converges locally uniformly to $f=0$. Neverthelessfor every $n$. Thus the proposed global conclusion is false.

A minimizing sequence in ${\mathrm{Lip}}_g(\Omega ,k)$ is uniformly bounded and equi-Lipschitz. The Arzela-Ascoli theorem gives a uniformly convergent subsequence with limit $u$ having the same boundary data and Lipschitz constant at most $k$. Its gradients have a weak-star convergent subsequence in $L^{\infty }$, with limit $Du$. Convexity of $F$ makes the integral functional weak-star lower semicontinuous, soThus $u$ attains the infimum by the direct method in the calculus of variations.

Subtracting the quadratic Taylor polynomial at the rescaling centre makes the function, gradient and Hessian matrix vanish there. Dividing by $A{r}^{2+\mu }$ preserves the normalized Hölder oscillation of the Hessian matrix while turning a small forcing-to-oscillation ratio into a harmonic limiting equation. The three vanished Taylor coefficients provide local bounds needed by the Arzela-Ascoli theorem.