Arzel?-Ascoli is compactness for families of functions. It says that uniform boundedness and uniform continuity across a family produce convergent subsequences.
Equicontinuity
Equicontinuity asks for one continuity scale that works for the whole family.
Definition. A family \(\mathcal F\subset C(K)\) is equicontinuous if for every \(\varepsilon>0\) there is \(\delta>0\) such that \(d(x,y)\lt \delta\Rightarrow |f(x)-f(y)|\lt \varepsilon\) for every \(f\in\mathcal F\).
Example. A family of Lipschitz functions with a common Lipschitz constant is equicontinuous; no bound on their absolute levels is needed for this conclusion. Conversely, the uniformly bounded family \(f_n(x)=x^n\) on \([0,1]\) is not equicontinuous at \(1\): with \(x_n=1-1/n\), \[ |f_n(1)-f_n(x_n)|=1-(1-1/n)^n\longrightarrow1-e^{-1}. \]
Finite nets and oscillation control
Compactness of the domain reduces uniform control to finitely many points.
Lemma. If \(K\) is compact and \(\mathcal F\) is equicontinuous, then for every \(\varepsilon>0\) there are points \(x_1,\ldots,x_m\in K\) such that every \(x\in K\) is within a \(\delta\)-ball of some \(x_i\), and \(|f(x)-f(x_i)|\lt \varepsilon\) for all \(f\in\mathcal F\).
Proof. Choose \(\delta\) from equicontinuity for \(\varepsilon\). Compactness gives a finite cover of \(K\) by \(\delta\)-balls centered at points \(x_i\). The stated oscillation bound is exactly the equicontinuity estimate.
Arzel?-Ascoli theorem
Pointwise subsequential compactness plus equicontinuity gives uniform subsequential compactness.
Theorem (Arzel?-Ascoli). Let \(K\) be compact metric. If \(\mathcal F\subset C(K)\) is pointwise bounded and equicontinuous, then every sequence in \(\mathcal F\) has a uniformly convergent subsequence. On compact \(K\), pointwise boundedness plus equicontinuity also implies uniform boundedness.
Proof. Compact metric spaces are separable: for each \(m\), choose a finite \(1/m\)-net and take the countable union of all net points. Enumerate a resulting dense set as \(D=\{q_1,q_2,\ldots\}\). Given \((f_n)\subset\mathcal F\), pointwise boundedness lets Bolzano-Weierstrass select a subsequence converging at \(q_1\); from it select one converging at \(q_2\), and so on. The diagonal subsequence \((g_n)\) converges at every \(q_j\).
Fix \(\varepsilon>0\). Equicontinuity supplies \(\delta>0\) such that \(d(x,y)<\delta\) implies \(|f(x)-f(y)|<\varepsilon/3\) for all \(f\in\mathcal F\). Choose a finite \(\delta\)-net \(q_{j_1},\ldots,q_{j_m}\) from \(D\). Because \(g_n(q_{j_r})\) converges for every \(r\), there is \(N\) such that \[ |g_n(q_{j_r})-g_\ell(q_{j_r})|<\varepsilon/3 \] for all \(n,\ell\ge N\) and all \(r\). For arbitrary \(x\), choose \(q_{j_r}\) within \(\delta\); a three-term estimate gives \(|g_n(x)-g_\ell(x)|<\varepsilon\). Thus \((g_n)\) is uniformly Cauchy and converges uniformly to a continuous function.
Finally, fix \(x_0\in K\). Pointwise boundedness gives \(\sup_{f\in\mathcal F}|f(x_0)|<\infty\). Equicontinuity and a finite chain of neighborhoods, or more directly a finite \(\delta\)-net with pointwise bounds at its centers, yield \(\sup_{f,x}|f(x)|<\infty\).
Compactness in \(C(K)\)
Arzel?-Ascoli characterizes relatively compact subsets of the sup-norm space.
Corollary. A subset of \(C(K)\), with \(K\) compact metric, is relatively compact in \(\|\cdot\|_\infty\) if and only if it is uniformly bounded and equicontinuous.
Proof. The reverse direction is the theorem. For necessity, let \(\overline{\mathcal F}\) be compact. It is bounded in the normed space \(C(K)\), so \(\mathcal F\) is uniformly bounded. Given \(\varepsilon>0\), choose a finite \(\varepsilon/3\)-net \(g_1,\ldots,g_m\) for \(\mathcal F\). Each \(g_j\) is uniformly continuous on compact \(K\); choose one \(\delta>0\) that works for all finitely many \(g_j\). For \(f\in\mathcal F\), select \(g_j\) with \(\|f-g_j\|_\infty<\varepsilon/3\). If \(d(x,y)<\delta\), then \[ |f(x)-f(y)| \le |f(x)-g_j(x)|+|g_j(x)-g_j(y)|+|g_j(y)-f(y)| <\varepsilon. \] Thus \(\mathcal F\) is equicontinuous.
Finite nets behind stochastic equicontinuity
The deterministic proof separates control at finitely many locations from control of oscillations between them.
Remark. The same architecture reappears for random functions: finite-dimensional convergence controls values on a finite net, while stochastic equicontinuity controls the gaps. This is the bridge to the tightness criteria in Empirical processes.