Multivariable Proof Workshop

A Workshop in Proof Architecture

The previous tutorial established that a continuous function on a nonempty compact subset of \(\mathbb{R}^n\) attains its maximum and minimum. This tutorial turns that existence result into a working proof method. We will follow sequences of points whose function values approach an optimum, and we will ask how much an optimum can change when the objective is altered slightly.

The recurring architecture is simple but powerful: choose points that capture the quantity of interest, use compactness to extract a convergent subsequence, and use continuity to identify the value at its limit. Each step has a distinct role. The choice of points translates a supremum or approximation into a sequence; compactness keeps a limit inside the domain; continuity connects the limiting function value to the limiting point.

We will use the Extreme Value Theorem in \(\mathbb{R}^n\) and the Compactness and Sequential Compactness in \(\mathbb{R}^n\) Theorem from earlier in this course. The aim is not to repeat those results, but to practice building new conclusions from them while keeping track of every hypothesis.

Maximizers and Sequences Approaching the Maximum

The Extreme Value Theorem ensures that \(A\) is nonempty. A sequence \((x_j)\) in \(K\) is called a maximizing sequence if \(f(x_j)\to M\). Its points need not themselves be maximizers. The important question is what compactness and continuity force about the sequence’s possible limit points.

Proof. Let \((x_{j_\ell})\) be a convergent subsequence, and write \(x_{j_\ell}\to a\). Since all its terms belong to \(K\), compactness and sequential compactness imply that its limit \(a\) belongs to \(K\). Continuity of \(f\) at \(a\) gives \(f(x_{j_\ell})\to f(a)\). On the other hand, \(f(x_j)\to M\), so the subsequence of function values also tends to \(M\). Uniqueness of limits in \(\mathbb{R}\) yields \(f(a)=M\). Therefore \(a\in A\).

For the final assertion, apply sequential compactness directly to the sequence \((x_j)\) in \(K\). It has a convergent subsequence; the argument just given shows that the subsequence’s limit is a maximizer. \(\square\)

The theorem also gives a useful stronger description: a maximizing sequence cannot stay a fixed positive distance away from all maximizers. To make that statement precise, define the distance from a point \(x\) to the nonempty set \(A\) by \(\operatorname{dist}(x,A)=\inf_{a\in A}\|x-a\|_2\). For every maximizing sequence, \(\operatorname{dist}(x_j,A)\to0\). Indeed, if this failed, there would be some \(\varepsilon>0\) and a subsequence with \(\operatorname{dist}(x_{j_\ell},A)\geq\varepsilon\) for every \(\ell\). Compactness would give a further subsequence converging to \(a\in A\), by the theorem. But then \(\operatorname{dist}(x_{j_\ell},A)\leq\|x_{j_\ell}-a\|_2\to0\), a contradiction.

Stability of the Optimal Value

In applications, an objective may be approximated by a simpler function, or its values may contain small measurement or computational errors. A useful proof question is then: if two objectives are uniformly close on the same compact domain, how different can their maximum values be? Uniform closeness means that a single error bound works at every point of the domain.

Proof. Let \(M_f=\max_{x\in K}f(x)\) and \(M_g=\max_{x\in K}g(x)\). By the Extreme Value Theorem, choose \(a,b\in K\) with \(f(a)=M_f\) and \(g(b)=M_g\). At \(a\), the pointwise bound gives \(f(a)\leq g(a)+\eta\), and \(g(a)\leq M_g\). Therefore \(M_f\leq M_g+\eta\). At \(b\), we have \(g(b)\leq f(b)+\eta\), and \(f(b)\leq M_f\), so \(M_g\leq M_f+\eta\). Together these inequalities imply \(|M_f-M_g|\leq\eta\).

For minima, apply the maximum statement to the continuous functions \(-f\) and \(-g\). Their maximum values are \(-\min_K f\) and \(-\min_K g\). Hence

This proof uses actual maximizing points, whose existence comes from compactness and continuity. The pointwise error bound is then evaluated at those points. Notice that the result controls the optimal values; it does not claim that the locations of the maximizers are close. Maximizers can move substantially, especially when the original objective has several nearly optimal points. \(\square\)

Where the Proof Can Fail

Compactness is not decorative in these arguments. It prevents sequences of nearly optimal points from escaping the domain without a convergent subsequence. Continuity is equally essential: it is what allows a limit of points to determine the limiting function value. When either link is missing, the conclusions may fail.

A reliable multivariable proof should make its dependencies visible. Before extracting a subsequence, identify the compact set containing the sequence. Before passing a function value to a limit, verify continuity at the limit point. Before asserting attainment, check that the domain is nonempty and that the Extreme Value Theorem applies. These checks are especially important when the domain is a constraint set: closedness relative to an open domain does not automatically mean closedness in \(\mathbb{R}^n\).

These techniques apply beyond optimization. In many multivariable arguments, a contradiction produces a sequence, compactness extracts a convergent subsequence, and continuity or differentiability identifies the limit. The proof becomes easier to audit when each of those moves is stated explicitly and the conclusion is no stronger than the argument establishes.