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Chapter 6: Problem 9

Let \(f_{n}:[0,1] \rightarrow \mathbb{R}\) be a sequence of increasing functions (that is, \(f_{n}(x) \geq f_{n}(y)\) whenever \(x \geq y) .\) Suppose \(f_{n}(0)=0\) and \(\lim _{n \rightarrow \infty} f_{n}(1)=0 .\) Show that \(\left\\{f_{n}\right\\}\) converges uniformly to \(0 .\)

Short Answer

Expert verified
The sequence \\(\{f_{n}\}\\) converges uniformly to 0 on \\([0,1]\\).

Step by step solution

01

Understand Uniform Convergence

A sequence of functions \(\{ f_n \}\) converges uniformly to a function \(f\) on \([a, b]\) if, for every \(\epsilon > 0\), there exists an \(N\) such that for all \(n > N\) and all \(x \in [a, b]\), \(|f_n(x) - f(x)| < \epsilon\). In this exercise, we aim to show that \(f_n(x)\) converges uniformly to \(0\) on \([0,1]\).
02

Evaluate Pointwise Convergence

We know that by hypothesis, \(f_n(1) ightarrow 0\). Since each \(f_n\) is an increasing function with \(f_n(0) = 0\), for any \(x \in [0, 1]\), \(0 \leq f_n(x) \leq f_n(1)\). This means that \(f_n(x) ightarrow 0\) for each \(x\), so \(f_n(x)\) converges pointwise to \(0\) on \([0, 1]\).
03

Show Uniform Convergence Using an Inequality

Given \(\epsilon > 0\), since \(f_n(1) ightarrow 0\), there exists \(N\) such that for all \(n > N\), \(|f_n(1) - 0| < \epsilon\). For increasing functions, \(f_n(x) \leq f_n(1)\) for all \(x \in [0, 1]\), therefore \( |f_n(x) - 0| = f_n(x) < \epsilon\). This satisfies the condition for uniform convergence.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Increasing Functions
An increasing function is one where as the input value increases, the output value does not decrease. In more formal mathematical terms, we say that a function \(f\) is increasing on an interval \([a, b]\) if for any two points \(x\) and \(y\) in that interval, if \(x \leq y\), then \(f(x) \leq f(y)\). This property ensures that the graph of the function does not descend.
  • **Monotonicity:** The core characteristic of increasing functions is monotonicity. In this context, a function that does not decrease at all is specifically referred to as non-decreasing or weakly increasing.
  • **Relevance in Convergence:** In the exercise provided, the sequence of functions \(\{f_n\}\) is increasing, making it easier to evaluate convergence properties like pointwise and uniform convergence. These functions start from \(f_n(0)=0\) and eventually converge to zero at \(f_n(1)=0\).
The increasing nature of the functions implies that they are bounded between zero and their value at one, facilitating a systematic evaluation of their convergence.
Pointwise Convergence
Pointwise convergence is when each point \(x\) in the domain, the sequence of functions \(\{f_n(x)\}\) converges to a function \(f(x)\). In simpler terms, as \(n\) increases, the value of each function in the sequence approaches the value of some limiting function \(f(x)\) for every \(x\) individually.
  • **Application to the Exercise:** For the given sequence \(\{f_n\}\), observe that each function \(f_n\) is increasing with \(f_n(0) = 0\). Therefore, as \(n\) grows, each function value at \(x\), \(f_n(x)\), must also approach zero because \(f_n(1) \to 0\).
  • **Evaluation Across Domain:** Importantly, because \(0 \leq f_n(x) \leq f_n(1)\) for all \(x \in [0,1]\), every point \(x\) sees its corresponding sequence of function values \(f_n(x)\) closing in on zero, confirming pointwise convergence across the entire interval.
Pointwise convergence is the initial step to understanding how functions behave, and it lays the groundwork for discussing more stringent forms, like uniform convergence.
Real Analysis
Real analysis is a branch of mathematics that deals with real numbers and real-valued sequences and functions. It is fundamental for understanding concepts like continuity, convergence, and limits in real-numbered contexts.
  • **Importance of Real Analysis:** In the context of the exercise, real analysis tools are crucial for proving properties such as pointwise and uniform convergence. Notably, understanding these convergence types helps learn about the behavior of function sequences.
  • **Analyzing Convergence in Real Analysis:** Real analysis offers a rigorous methodology for discussing when and how functions and sequences "settle down" to a steady state or value. This is pivotal in studying uniform convergence, where over the entire interval, a function sequence converges to a single function exceedingly closely.
Real analysis not only enhances insights into the behavior of sequences but is central in ensuring mathematical precision. This exercise is an excellent example of applying real analysis principles to establish uniform convergence from initial pointwise convergence insights.

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Most popular questions from this chapter

While uniform convergence preserves continuity, it does not preserve differentiability. Find an explicit example of a sequence of differentiable functions on [-1,1] that converge uniformly to a function \(f\) such that \(f\) is not differentiable. Hint: There are many possibilities, simplest is perhaps to combine \(|x|\) and \(\frac{n}{2} x^{2}+\frac{1}{2 n},\) another is to consider \(\sqrt{x^{2}+(1 / n)^{2}}\). Show that these functions are differentiable, converge uniformly, and then show that the limit is not differentiable.

A joint limit (see above) does not mean the iterated limits even exist. Consider \(x_{n, m}:=\) \(\frac{(-1)^{n+m}}{\min [n, m]}\) a) Show that for no \(n\) does \(\lim _{m \rightarrow \infty} x_{n, m}\) exist, and for no \(m\) does \(\lim _{n \rightarrow \infty} x_{n, m}\) exist. So neither \(\lim _{n \rightarrow \infty} \lim _{n \rightarrow \infty} x_{n, m}\) nor \(\lim _{m \rightarrow \infty} \lim _{n \rightarrow \infty} x_{n, m}\) makes any sense at all. b) Show that the joint limit of \(\left\\{x_{n-m}\right\\}\) exists and is \(0 .\)

Let \(\left\\{f_{n}\right\\}\) be a sequence of functions defined on \([0,1] .\) Suppose there exists a sequence of distinct numbers \(x_{n} \in[0,1]\) such that $$ f_{n}\left(x_{n}\right)=1 $$ Prove or disprove the following statements: a) True or false: There exists \(\left\\{f_{n}\right\\}\) as above that converges to 0 pointwise. b) True or false: There exists \(\left\\{f_{n}\right\\}\) as above that converges to 0 uniformly on \([0,1] .\)

The Dirichlet function \(f:[0,1] \rightarrow \mathbb{R}\), that is the function such that \(f(x):=1\) if \(x \in \mathbb{Q}\) and \(f(x):=0\) if \(x \notin Q\), is not the pointwise limit of continuous functions, although this is difficult to show. Prove, however, that \(f\) is a pointwise limit of functions that are themselves pointwise limits of continuous functions themselves.

Let \(\left\\{f_{n}\right\\},\left\\{g_{n}\right\\}\) and \(\left\\{h_{n}\right\\}\) be sequences of functions on \([a, b] .\) Suppose \(\left\\{f_{n}\right\\}\) and \(\left\\{h_{n}\right\\}\) converge uniformly to some function \(f:[a, b] \rightarrow \mathbb{R}\) and suppose \(f_{n}(x) \leq g_{n}(x) \leq h_{n}(x)\) for all \(x \in[a, b] .\) Show that \(\left\\{g_{n}\right\\}\) converges uniformly to \(f\).
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