Arzelà's Bounded Convergence Theorem A Detailed Proof And Applications

In the realm of real analysis, Arzelà's Bounded Convergence Theorem stands as a cornerstone for understanding the convergence of sequences of functions. This theorem, a powerful extension of the Dominated Convergence Theorem, provides conditions under which the limit of the integral of a sequence of functions equals the integral of the limit function. Specifically, it addresses scenarios where the pointwise limit exists and the sequence of functions is uniformly bounded and equicontinuous. Understanding this theorem is crucial for advanced calculus, functional analysis, and various applications in mathematical physics and engineering.

The significance of Arzelà's theorem lies in its ability to handle sequences of functions that may not have a dominating integrable function, a requirement in the more restrictive Dominated Convergence Theorem. By focusing on uniform boundedness and equicontinuity, Arzelà's theorem carves out a broader class of functions for which convergence results can be rigorously established. This makes it an indispensable tool in many areas of mathematical analysis.

This article delves into a specific proof of Arzelà's Bounded Convergence Theorem as presented in the first edition of Tom M. Apostol's renowned Mathematical Analysis. Apostol's exposition offers a clear and insightful approach to understanding the theorem's intricacies. We will dissect the theorem's statement, meticulously examine the assumptions, and provide a detailed walkthrough of Apostol's proof. This exploration will not only enhance your understanding of the theorem itself but also provide a deeper appreciation for the techniques and arguments commonly employed in real analysis.

Theorem Statement

Before diving into the proof, let's formally state Arzelà's Bounded Convergence Theorem as it appears in Apostol's Mathematical Analysis. This will serve as our roadmap, guiding us through the logical steps and ensuring we fully grasp the theorem's conditions and conclusion.

Theorem 13-17 (Arzelà). Assume that $\{f_{n}\rbrace$ is a uniformly bounded sequence of real-valued functions defined on a closed and bounded interval $[a, b]$. Assume also that $\{f_{n}\rbrace$ converges pointwise on $[a, b]$ to a function $f$, and that each $f_{n}$ is Riemann-integrable on $[a, b]$. If $f$ is also Riemann-integrable on $[a, b]$, then

$$\lim_{n \to \infty} \int_{a}^{b} f_{n}(x) dx = \int_{a}^{b} f(x) dx.$$

To fully appreciate the theorem, let's unpack its key components:

• Uniformly Bounded Sequence: A sequence of functions $\{f_{n}\rbrace$ is uniformly bounded on $[a, b]$ if there exists a constant $M > 0$ such that $|f_{n}(x)| \leq M$ for all $n$ and for all $x \in [a, b]$. In simpler terms, all functions in the sequence are bounded by the same constant $M$ across the entire interval.
• Pointwise Convergence: A sequence of functions $\{f_{n}\rbrace$ converges pointwise to $f$ on $[a, b]$ if for each $x \in [a, b]$, $\lim_{n \to \infty} f_{n}(x) = f(x)$. This means that for any fixed $x$, the sequence of function values converges to the function value at $x$.
• Riemann-Integrability: Each $f_{n}$ and the limit function $f$ are Riemann-integrable on $[a, b]$. This condition ensures that the integrals $\int_{a}^{b} f_{n}(x) dx$ and $\int_{a}^{b} f(x) dx$ are well-defined.

The theorem's conclusion states that under these conditions, we can interchange the limit and the integral. That is, the limit of the integrals of the $f_{n}$ is equal to the integral of the pointwise limit function $f$. This is a powerful result, but it's crucial to recognize that it doesn't hold for all sequences of functions. The conditions of uniform boundedness and Riemann-integrability are essential for its validity.

In the following sections, we will delve into Apostol's proof, dissecting each step and providing detailed explanations to solidify your understanding. We will see how these conditions are strategically used to establish the convergence result.

Proof of Arzelà's Bounded Convergence Theorem

Now, let's embark on a detailed exploration of Apostol's proof of Arzelà's Bounded Convergence Theorem. We'll break down the proof into manageable steps, providing explanations and insights along the way. This approach will allow you to grasp the logic behind each step and appreciate the elegance of the overall argument.

Apostol's proof leverages the properties of Riemann integrals, uniform boundedness, and pointwise convergence to demonstrate the interchangeability of the limit and the integral. It cleverly uses the definition of the Riemann integral and carefully constructs inequalities to arrive at the desired conclusion. The proof is a beautiful example of the techniques used in real analysis to establish fundamental convergence results.

To begin, we restate the theorem for clarity:

Theorem: Assume that $\{f_{n}\rbrace$ is a uniformly bounded sequence of real-valued functions defined on a closed and bounded interval $[a, b]$. Assume also that $\{f_{n}\rbrace$ converges pointwise on $[a, b]$ to a function $f$, and that each $f_{n}$ is Riemann-integrable on $[a, b]$. If $f$ is also Riemann-integrable on $[a, b]$, then

$$\lim_{n \to \infty} \int_{a}^{b} f_{n}(x) dx = \int_{a}^{b} f(x) dx.$$

Proof:

1. Setting the Stage: The Goal

   Our goal is to show that for any given $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n > N$,

   $$\left| \int_{a}^{b} f_{n}(x) dx - \int_{a}^{b} f(x) dx \right| < \epsilon.$$

   This inequality precisely captures the notion that the limit of the integrals of $f_{n}$ is equal to the integral of $f$.

2. Exploiting Riemann Integrability

   Since $f$ is Riemann-integrable on $[a, b]$, for any $\epsilon > 0$, there exists a partition $P = \{x_{0}, x_{1}, ..., x_{k}\}$ of $[a, b]$ such that the upper and lower Riemann sums of $f$ satisfy

   $$U(P, f) - L(P, f) < \frac{\epsilon}{2}.$$

   This is a fundamental property of Riemann-integrable functions: the difference between the upper and lower sums can be made arbitrarily small by choosing a sufficiently fine partition. This partition will be crucial in controlling the error in our approximation.

3. Leveraging Uniform Boundedness

   Because $\{f_{n}\rbrace$ is uniformly bounded, there exists a constant $M > 0$ such that $|f_{n}(x)| \leq M$ for all $n$ and for all $x \in [a, b]$. This uniform bound will help us control the size of the functions and their integrals.

4. Harnessing Pointwise Convergence

   Since $f_{n}$ converges pointwise to $f$ on $[a, b]$, for each $x \in [a, b]$, we have $\lim_{n \to \infty} f_{n}(x) = f(x)$. This means that for each $x_{i}$ in our partition $P$, there exists an $N_{i}$ such that for all $n > N_{i}$,

   $$|f_{n}(x_{i}) - f(x_{i})| < \frac{\epsilon}{4k(b - a)},$$

   where $k$ is the number of subintervals in the partition $P$. This inequality is the heart of the pointwise convergence condition, ensuring that the function values get arbitrarily close at each point in the partition.

5. Choosing a Suitable N

   Let $N = \max\{N_{1}, N_{2}, ..., N_{k}\}$. Then, for all $n > N$, we have

   $$|f_{n}(x_{i}) - f(x_{i})| < \frac{\epsilon}{4k(b - a)}$$

   for all $i = 0, 1, ..., k$. This choice of $N$ ensures that the pointwise convergence condition holds simultaneously for all points in the partition.

6. Constructing the Integral Difference

   Now, for $n > N$, consider the difference

   $$\left| \int_{a}^{b} f_{n}(x) dx - \int_{a}^{b} f(x) dx \right| = \left| \int_{a}^{b} (f_{n}(x) - f(x)) dx \right| \leq \int_{a}^{b} |f_{n}(x) - f(x)| dx.$$

   This step uses the triangle inequality for integrals, a fundamental property that allows us to bound the absolute value of the integral by the integral of the absolute value.

7. Bounding the Integral Difference

   We now need to carefully bound the integral $\int_{a}^{b} |f_{n}(x) - f(x)| dx$. To do this, we divide the interval $[a, b]$ into the subintervals defined by our partition $P$:

   $$\int_{a}^{b} |f_{n}(x) - f(x)| dx = \sum_{i=1}^{k} \int_{x_{i-1}}^{x_{i}} |f_{n}(x) - f(x)| dx.$$

   This decomposition allows us to analyze the integral piece by piece.

8. Bounding within Subintervals

   For each subinterval $[x_{i-1}, x_{i}]$, we use the fact that $f_{n}$ and $f$ are Riemann-integrable. Let $M_{i}$ and $m_{i}$ be the supremum and infimum of $f(x)$ on $[x_{i-1}, x_{i}]$, respectively, and let $M_{i,n}$ and $m_{i,n}$ be the supremum and infimum of $f_{n}(x)$ on $[x_{i-1}, x_{i}]$, respectively. Then,

   $$\int_{x_{i-1}}^{x_{i}} |f_{n}(x) - f(x)| dx \leq (M_{i,n} - m_{i,n}) (x_{i} - x_{i-1}).$$

   This inequality bounds the integral over the subinterval by the difference between the supremum and infimum of $|f_{n}(x) - f(x)|$ multiplied by the length of the subinterval.

9. Using Pointwise Convergence Again

   We now use the pointwise convergence condition to bound $M_{i,n} - m_{i,n}$. For $n > N$, we have

   $$M_{i,n} - m_{i,n} \leq (M_{i} - m_{i}) + \frac{\epsilon}{2k(b - a)}.$$

   This inequality cleverly relates the difference between the supremum and infimum of $f_{n}(x)$ to the difference between the supremum and infimum of $f(x)$ plus a small error term. The error term comes from the pointwise convergence condition.

10. Putting it All Together

    Substituting this bound back into our sum, we get

    $$\int_{a}^{b} |f_{n}(x) - f(x)| dx \leq \sum_{i=1}^{k} \left[ (M_{i} - m_{i}) + \frac{\epsilon}{2k(b - a)} \right] (x_{i} - x_{i-1}).$$

    Expanding the sum, we have

    $$\int_{a}^{b} |f_{n}(x) - f(x)| dx \leq \sum_{i=1}^{k} (M_{i} - m_{i}) (x_{i} - x_{i-1}) + \sum_{i=1}^{k} \frac{\epsilon}{2k(b - a)} (x_{i} - x_{i-1}).$$

    The first term is simply the difference between the upper and lower sums of $f$ for the partition $P$, which we know is less than $\epsilon/2$. The second term simplifies to $\epsilon/2$:

    $$\int_{a}^{b} |f_{n}(x) - f(x)| dx \leq U(P, f) - L(P, f) + \frac{\epsilon}{2} < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$$

11. The Grand Finale

    Therefore, for all $n > N$,

    $$\left| \int_{a}^{b} f_{n}(x) dx - \int_{a}^{b} f(x) dx \right| < \epsilon.$$

    This completes the proof. We have shown that the limit of the integrals of $f_{n}$ is indeed equal to the integral of $f$, under the conditions of Arzelà's Bounded Convergence Theorem.

Significance of the Proof

Apostol's proof elegantly demonstrates how the conditions of uniform boundedness, pointwise convergence, and Riemann-integrability work together to ensure the interchangeability of limits and integrals. The careful construction of inequalities and the strategic use of the definition of the Riemann integral highlight the power of real analysis techniques in establishing fundamental convergence results. This proof is not only a valuable tool for understanding Arzelà's theorem but also a testament to the beauty and rigor of mathematical reasoning.

Applications and Examples

Arzelà's Bounded Convergence Theorem is not merely a theoretical result; it has significant applications in various areas of mathematics and its related fields. Its power lies in providing conditions under which we can interchange limits and integrals, a crucial operation in many analytical arguments. Let's explore some key applications and examples to illustrate its utility.

1. Justifying Differentiation Under the Integral Sign

One of the most important applications of Arzelà's theorem is in justifying differentiation under the integral sign. This technique allows us to compute the derivative of an integral with respect to a parameter by differentiating the integrand directly. Suppose we have an integral of the form

$$F(t) = \int_{a}^{b} f(x, t) dx,$$

and we want to find $F'(t)$. The natural approach is to differentiate under the integral sign:

$$F'(t) = \int_{a}^{b} \frac{\partial}{\partial t} f(x, t) dx.$$

However, this operation is not always valid. Arzelà's theorem provides a rigorous justification under certain conditions. If the partial derivative $\frac{\partial}{\partial t} f(x, t)$ exists and is continuous, and if the sequence of difference quotients

$$\frac{f(x, t + h_{n}) - f(x, t)}{h_{n}}$$

is uniformly bounded for some sequence $h_{n} \to 0$, then we can apply Arzelà's theorem to conclude that differentiation under the integral sign is valid.

2. Solving Differential Equations

Arzelà's theorem plays a vital role in establishing the existence and uniqueness of solutions to certain differential equations. For instance, in the Picard-Lindelöf theorem, which guarantees the existence and uniqueness of solutions to initial value problems, Arzelà's theorem is used to show the convergence of a sequence of approximate solutions to the true solution. By carefully constructing a sequence of functions that satisfy the differential equation approximately and then applying Arzelà's theorem, we can prove that the limit function is indeed a solution.

3. Evaluating Limits of Integrals

In many situations, we encounter integrals whose limits are difficult to evaluate directly. Arzelà's theorem provides a powerful tool for computing such limits. Suppose we have a sequence of functions $f_{n}(x)$ that converges pointwise to $f(x)$, and we want to find

$$\lim_{n \to \infty} \int_{a}^{b} f_{n}(x) dx.$$

If the conditions of Arzelà's theorem are satisfied (uniform boundedness and Riemann-integrability), then we can simply evaluate the integral of the limit function:

$$\lim_{n \to \infty} \int_{a}^{b} f_{n}(x) dx = \int_{a}^{b} f(x) dx.$$

This can often simplify the problem significantly, as the limit function may be easier to integrate than the original sequence of functions.

4. Examples

Let's illustrate the application of Arzelà's theorem with a concrete example.

Example 1: Consider the sequence of functions $f_{n}(x) = \frac{\sin(nx)}{n}$ on the interval $[0, \pi]$. We want to find

$$\lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin(nx)}{n} dx.$$

First, note that $f_{n}(x)$ converges pointwise to $f(x) = 0$ on $[0, \pi]$. Also, $|f_{n}(x)| = \left| \frac{\sin(nx)}{n} \right| \leq \frac{1}{n} \leq 1$ for all $n$ and $x \in [0, \pi]$, so the sequence is uniformly bounded. Each $f_{n}(x)$ is continuous and therefore Riemann-integrable, and the limit function $f(x) = 0$ is also Riemann-integrable. Thus, all the conditions of Arzelà's theorem are satisfied. Therefore,

$$\lim_{n \to \infty} \int_{0}^{\pi} \frac{\sin(nx)}{n} dx = \int_{0}^{\pi} 0 dx = 0.$$

Example 2: Consider the sequence of functions $f_{n}(x) = x^{n}$ on the interval $[0, 1]$. This sequence converges pointwise to the function

$$f(x) = \begin{cases} 0, & 0 \leq x < 1 \\ 1, & x = 1 \end{cases}$$

Each $f_{n}(x)$ is continuous and Riemann-integrable, but the limit function $f(x)$ is also Riemann-integrable. However, the sequence is not uniformly bounded, so we cannot directly apply Arzelà's theorem. In this case, we can compute the limit directly:

$$\int_{0}^{1} x^{n} dx = \frac{1}{n + 1},$$

and

$$\lim_{n \to \infty} \int_{0}^{1} x^{n} dx = \lim_{n \to \infty} \frac{1}{n + 1} = 0.$$

On the other hand,

$$\int_{0}^{1} f(x) dx = \int_{0}^{1} \begin{cases} 0, & 0 \leq x < 1 \\ 1, & x = 1 \end{cases} dx = 0.$$

In this case, the limit of the integrals is equal to the integral of the limit, but this is not guaranteed in general when the conditions of Arzelà's theorem are not met.

These examples illustrate the power and limitations of Arzelà's Bounded Convergence Theorem. When the conditions are satisfied, it provides a powerful tool for interchanging limits and integrals. However, it's crucial to verify that the conditions are indeed met before applying the theorem.

Conclusion

In conclusion, Arzelà's Bounded Convergence Theorem is a fundamental result in real analysis that provides conditions for interchanging limits and integrals. This theorem, particularly in Apostol's exposition, offers a clear and insightful understanding of the theorem's intricacies. The key conditions—uniform boundedness, pointwise convergence, and Riemann-integrability—play crucial roles in ensuring the validity of this interchange. Apostol's proof, meticulously dissected in this article, elegantly demonstrates how these conditions interact to yield the desired convergence result. By understanding the proof, we gain a deeper appreciation for the techniques and arguments commonly employed in real analysis.

Furthermore, Arzelà's theorem has significant applications in various areas of mathematics and its related fields. From justifying differentiation under the integral sign to solving differential equations and evaluating limits of integrals, its utility is undeniable. The examples discussed highlight how the theorem can be applied in practice and also underscore the importance of verifying the conditions before applying the theorem.

Arzelà's Bounded Convergence Theorem stands as a testament to the rigor and elegance of real analysis. Its study not only enhances our understanding of convergence phenomena but also equips us with powerful tools for tackling a wide range of analytical problems. Whether you are a student delving into the intricacies of real analysis or a seasoned researcher applying these concepts in advanced studies, Arzelà's theorem remains an indispensable part of your mathematical toolkit. Its enduring significance lies in its ability to bridge the gap between limits and integrals, providing a solid foundation for many advanced mathematical techniques.