Question

Prove (without using Theorem 4.4 .10\():\) If each \(F_{n}\) is nondecreasing and \(\left\\{F_{n}\right\\}\) converges uniformly to \(F\) on \([a, b],\) then $$ \lim _{n \rightarrow \infty} \int_{a}^{b} F_{n}(x) d x=\int_{a}^{b} F(x) d x. $$

Short answer

Question: Prove that if a sequence of functions \(F_n\) is nondecreasing on the interval \([a,b]\) and converges uniformly to a function \(F\) on \([a,b]\), then the limit of the integral of \(F_n\) as \(n\) approaches infinity is equal to the integral of \(F\) on \([a,b]\). Answer: The result can be proven using proof by contradiction, invoking uniform convergence, and manipulating inequalities involving integrals. By showing that assuming the limit of the integral of \(F_n\) as \(n\) approaches infinity is not equal to the integral of \(F\) leads to a contradiction, we can conclude that the limit of the integral of \(F_n\) as \(n\) approaches infinity is indeed equal to the integral of \(F\) on \([a,b]\).

Step-by-step solution

Proof by contradiction

Suppose that the limit of the integral of \(F_n\) as \(n\) approaches infinity is not equal to the integral of \(F\). That is, $$ \lim _{n \rightarrow \infty} \int_{a}^{b} F_{n}(x) dx \neq \int_{a}^{b} F(x) dx. $$ This means that there exists some \(\epsilon > 0\) such that for all natural numbers \(N\), there is some \(n \geq N\) with $$ \left|\int_{a}^{b} F_{n}(x) dx - \int_{a}^{b} F(x) dx\right| \geq \epsilon. $$

Invoke uniform convergence

Since \(F_n\) converges uniformly to \(F\) on \([a,b]\), for every \(\epsilon > 0\), there exists a natural number \(N_1\) such that for all \(n \geq N_1\) and for all \(x \in [a,b]\), $$ |F_n(x) - F(x)| < \frac{\epsilon}{2(b-a)}. $$

Use the properties of integrals to manipulate the inequality

Since our goal is to find a contradiction with the inequality from step 1 in terms of integrals we can rewrite the inequality as follows for all \(n \geq N_1\) : $$ \int_{a}^{b} |F_n(x) - F(x)| dx < \frac{\epsilon}{2}. $$

Apply the inequality in step 1

Choose \(n \geq N_1\) such that we have $$ \left|\int_{a}^{b} F_{n}(x) dx - \int_{a}^{b} F(x) dx\right| \geq \epsilon. $$ Now, we have $$ \left|\int_{a}^{b} F_{n}(x) dx - \int_{a}^{b} F(x) dx\right| = \left|\int_{a}^{b} (F_n(x) - F(x)) dx\right|. $$ Since the functions are nondecreasing, the subtraction inside the integral is positive, which leads us to $$ \int_{a}^{b} (F_n(x) - F(x)) dx \geq \epsilon. $$ Taking the absolute value, we have $$ \int_{a}^{b} |F_n(x) - F(x)| dx \geq \epsilon. $$

Show a contradiction

As we can see, the result we derived in step 4 contradicts the result derived in step 3, where we have $$ \int_{a}^{b} |F_n(x) - F(x)| dx < \frac{\epsilon}{2}. $$ Therefore, our initial assumption that the limit of the integral of \(F_n\) as \(n\) approaches infinity is not equal to the integral of \(F\) must be incorrect, and we can conclude that $$ \lim _{n \rightarrow \infty} \int_{a}^{b} F_{n}(x) dx=\int_{a}^{b} F(x) dx. $$

Key concepts

Real Analysis

Real Analysis is a branch of mathematics dealing with real numbers and the real-valued functions of a real variable. In addressing issues such as convergence, continuity, and integration, this field forms the theoretical underpinnings of calculus and provides a rigorous framework for applications across sciences and engineering. The topic of uniform convergence, which measures the convergence of a sequence of functions to a limiting function uniformly across a range, is a central concept in Real Analysis.

Understanding uniform convergence is critical in ensuring the passage of limits under the integral sign, which involves taking the limit of an integral or integrating a limit of functions. This technique is essential in analyzing the behavior of complex systems in physics and other sciences, where functions model real-world phenomena, and their limits symbolize long-term behavior or stability.

Nondecreasing Function

A function is considered to be nondecreasing on an interval if, for any two numbers in the interval, a larger input never produces a smaller output. In mathematical terms, a function \(f\) is nondecreasing on an interval \([a,b]\) if for every \(x_1\) and \(x_2\) in \([a,b]\) such that \(x_1 \<= x_2\), it follows that \(f(x_1) \<= f(x_2)\).

Nondecreasing functions play a vital role in real analysis because they preserve order and facilitate the proof of various integral properties, such as the one being explored in our exercise. When dealing with sequences of nondecreasing functions that converge to a limit function, the analysis of their behaviors becomes more tractable, particularly when discussing an integral's limit.

Integral Convergence

Integral convergence concerns itself with the behavior of integrals of sequences of functions as the sequence converges to some limit. It explores conditions under which the integral of the limit function equals the limit of integrals, a crucial aspect when it comes to applications in physics, economics, and other disciplines that utilize integration to model cumulative or continuous processes.

The convergence of integrals is not always guaranteed, which necessitates the establishment of conditions such as uniform convergence and properties like the nondecreasing nature of functions to affirm the exchange of limit processes with integration. This concept is at the heart of the exercise under discussion, which aims to demonstrate that the integrals of a sequence of nondecreasing functions converging uniformly will also converge to the integral of the limit function.

Proof by Contradiction

Proof by contradiction is a powerful logical method used in mathematics to establish the truth of a proposition. The technique involves assuming that the proposition is false and demonstrating that this assumption leads to a contradiction. The existence of the contradiction implies that the original assumption was incorrect, and therefore, the proposition must be true.

In the context of the given exercise, this method is invoked to prove integral convergence by assuming the opposite—and then showing that this assumption is incompatible with the known conditions of uniform convergence of the nondecreasing functions \(F_n\). This type of reasoning not only strengthens the understanding of students but also equips them with problem-solving skills that are effective in various mathematical and scientific fields.