Question

Suppose that \(\left\\{F_{n}\right\\}\) converges pointwise on \([a, b]\) and, for each \(x\) in \([a, b],\) there is an open interval \(I_{x}\) containing \(x\) such that \(\left\\{F_{n}\right\\}\) converges uniformly on \(I_{x} \cap[a, b]\). Show that \(\left\\{F_{n}\right\\}\) converges uniformly on \([a, b]\).

Short answer

Question: Prove that the sequence of functions converges uniformly on the interval [a, b] given that it converges pointwise on [a, b] and each point x ∈ [a, b] is contained in an open interval Ix where the sequence converges uniformly. Answer: To show uniform convergence, we can follow these steps: 1. Show that the family of open intervals {Ix: x ∈ [a, b]} forms an open cover for [a, b]. 2. Apply the Heine-Borel theorem to find a finite subcover. 3. Establish a bound on the convergence for the finite subcover. 4. Find the maximum index among the positive integers in the bound. 5. With the bound and the maximum index, prove uniform convergence on the entire interval [a, b].

Step-by-step solution

Show that the family of open intervals \(\left\\{I_{x}: x \in [a, b]\right\\}\) forms an open cover for \([a, b]\).

We need to show that the union of all the open intervals \(I_{x}\) for every point \(x \in [a, b]\) covers the entire interval \([a, b]\). By definition of an open interval containing \(x\), every point \(x \in [a, b]\) is an element of some \(I_{x}\). Therefore, the union of all \(I_{x}\) contains every point in \([a, b]\), which means \(\bigcup_{x\in[a, b]} I_x \supseteq [a, b]\).

Apply the Heine-Borel theorem

Since \([a, b]\) is compact, by the Heine-Borel theorem, there exists a finite subcover of \(\left\\{I_{x}: x \in [a, b]\right\\}\). Hence, we can find a finite collection of open intervals \(\left\\{I_{x_1},I_{x_2}, \cdots, I_{x_k}\right\\}\) such that \([a, b] \subseteq \bigcup_{i=1}^k I_{x_i}\).

Establish a bound on the convergence

For every interval \(I_{x_i}\), we are given that the sequence of functions converges uniformly on each \(I_{x_i} \cap [a, b]\). Therefore, for all \(x \in I_{x_i}\cap [a, b]\) and for every positive integer N, there exists a positive integer \(N_i\) such that $$ \left|F_n(x)-F(x)\right|<\frac{1}{N} $$ for all \(n \geq N_i\).

Find the maximum index

Since we have a finite subcover, we can select the maximum index among all the positive integers \(N_1, N_2, \cdots, N_k\). Let \(N^* = \max\left\\{N_1, N_2, \cdots, N_k\right\\}\). Now, for any \(x \in [a, b]\) and any \(n \geq N^*\), there must exist an interval \(I_{x_i}\) containing \(x\). Thus, we have $$ \left|F_n(x) - F(x)\right| < \frac{1}{N}. $$

Prove uniform convergence

With the bound found in Step 4, we have shown that for every point \(x \in [a, b]\) and \(n \geq N^*\), the difference between each term in the sequence and the limit is less than \(\frac{1}{N}\). As a result, for any given \(\epsilon>0\), there exists \(N^*\in \mathbb{N}\) such that $$ \left|F_n(x)-F(x)\right|<\epsilon $$ for all \(x \in [a, b]\) and \(n \geq N^*\). Therefore, the sequence of functions \(\left\\{F_{n}\right\\}\) converges uniformly on the interval \([a, b]\).

Key concepts

Pointwise Convergence

Pointwise convergence is a type of convergence that occurs within the broader context of real analysis. It involves a sequence of functions, and for each specific point in their domain, the function values get closer to a certain value as the sequence progresses.

To understand this concept in a practical sense, consider a sequence of functions \(\left\{ F_n\right\}\) defined on an interval \( [a, b] \). This sequence is said to converge pointwise if, for each point \( x \), the value of \( F_n(x)\) approaches a limiting value \( F(x) \) as \( n\) tends to infinity. In mathematical terms, we express this as: for every \( x \) in \( [a, b] \) and for every \( \epsilon > 0 \) there exists an integer \( N \) such that for all \( n \geq N \), it holds that \( \left | F_n(x) - F(x) \right | < \epsilon \).

The beauty of pointwise convergence lies in its simplicity; it allows one to focus on the behavior of the sequence at individual points. However, it does not provide information about the convergence behavior in aggregate across the entire domain, which is where uniform convergence comes into play—a topic that connects deeply with the Heine-Borel theorem as demonstrated in the exercise.

Heine-Borel Theorem

The Heine-Borel theorem plays a pivotal role in real analysis, especially in the study of uniform convergence. This theorem provides us with a connection between open covers and compact sets, which are vital concepts in topology.

An open cover for a set S is a collection of open sets whose union includes S. The Heine-Borel theorem states that a subset of Euclidean space is compact if and only if every open cover of that set has a finite subcover. In layman's terms, for any collection of open sets that cover a compact set, there are a limited number of these open sets that still cover the entire set.

In the context of the textbook exercise, the Heine-Borel theorem assures us that since \( [a, b] \) is a closed and bounded interval in the real numbers, it is compact. Given an open cover composed of intervals \( I_x \) where the sequence of functions converges uniformly, we can deduce the existence of a finite subcover. This finite collection of intervals becomes the cornerstone in establishing uniform convergence across the entire interval \( [a, b] \), bridging the gap between pointwise convergence at individual points and uniform convergence over the entire domain.

Sequence of Functions

In real analysis, a sequence of functions refers to an ordered set of functions where each function is mapped to a natural number. Exploring sequences of functions is a way of investigating the behavior of functions 'in the limit', as the sequence progresses to infinity.

For example, a sequence \( \left\{ F_n \right\} \) might represent different approximations to a more complex function or a process through time, such as heat distribution in a rod as it reaches a steady state.
Understanding the convergence of a sequence of functions is essential for various applications, such as solving differential equations and mathematical modeling. The behavior of such sequences can vary significantly; they might converge pointwise, uniformly, or not converge at all. Uniform convergence, the focus of our original exercise, is particularly powerful as it ensures that the limit of the sequence of functions behaves 'nicely', preserving properties such as continuity, integration, and differentiation.

Real Analysis

Real analysis is a branch of mathematics that deals with the behavior of real numbers, sequences and series of real numbers, and real-valued functions. Central to real analysis is the investigation of limits, continuity, differentiation, and integration.

Within real analysis, the concepts of various forms of convergence play critical roles. The distinction between pointwise and uniform convergence is particularly important because it has implications for the properties that might or might not be preserved as we pass to a limit of a sequence of functions. For instance, a pointwise limit of continuous functions need not be continuous itself, but a uniformly convergent sequence of continuous functions always has a continuous limit.

The exercise we're examining illustrates one of the many elegant connections in real analysis—between the convergence of functions and topological properties like compactness as characterized by the Heine-Borel theorem. It's essential for students to grasp these connections to appreciate the coherence and depth of the subject.