Real Analysis Via Sequences and Undergraduate Texts in Mathematics

Real analysis is a branch of mathematics that studies the real numbers and the functions that are defined on them. It is a fundamental subject for students of mathematics, science, and engineering, and it has applications in many fields, such as physics, economics, and computer science.

There are many different ways to approach real analysis, but one of the most common is to use sequences. A sequence is a list of numbers that are arranged in a specific order. Sequences can be used to represent a wide variety of mathematical objects, such as functions, graphs, and limits.

Real Analysis via Sequences and Series (Undergraduate Texts in Mathematics)

by Bruce van Brunt

4.4 out of 5

Language	:	English
File size	:	5918 KB
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Print length	:	487 pages

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In this article, we will explore some of the basic concepts of real analysis via sequences. We will begin by discussing the concept of convergence, and we will then move on to discuss limits, Cauchy sequences, and completeness. We will also briefly discuss metric spaces, continuity, derivatives, and integrals.

Convergence

One of the most important concepts in real analysis is convergence. A sequence is said to converge to a limit if the terms of the sequence get closer and closer to the limit as the index of the sequence increases.

There are many different ways to define convergence, but one of the most common is the epsilon-delta definition. This definition states that a sequence {x_n}converges to a limit L if for any number epsilon > 0, there exists a number delta > 0 such that |x_n - L| N, where N is some fixed natural number.

In other words, the epsilon-delta definition says that for any number epsilon > 0, we can find a number delta > 0 such that all of the terms of the sequence {x_n}that come after the N-th term are within epsilon of the limit L.

Convergence is a very important concept in real analysis, and it is used to define many other important concepts, such as limits, Cauchy sequences, and completeness.

Limits

A limit is a value that a sequence approaches as the index of the sequence increases. Limits can be used to find the value of a function at a point, to determine whether a function is continuous at a point, and to find the derivative of a function.

There are many different ways to find the limit of a sequence, but one of the most common is to use the epsilon-delta definition. This definition states that the limit of a sequence {x_n}is L if for any number epsilon > 0, there exists a number N such that |x_n - L| N.

In other words, the epsilon-delta definition says that for any number epsilon > 0, we can find a number N such that all of the terms of the sequence {x_n}that come after the N-th term are within epsilon of the limit L.

Limits are a very important concept in real analysis, and they are used to define many other important concepts, such as continuity, derivatives, and integrals.

Cauchy Sequences

A Cauchy sequence is a sequence that gets closer and closer to itself as the index of the sequence increases. Cauchy sequences are important because they can be used to prove that a sequence is convergent.

There are many different ways to define a Cauchy sequence, but one of the most common is the epsilon-delta definition. This definition states that a sequence {x_n}is Cauchy if for any number epsilon > 0, there exists a number N such that |x_m - x_n| N.

In other words, the epsilon-delta definition says that for any number epsilon > 0, we can find a number N such that all of the terms of the sequence {x_n}that come after the N-th term are within epsilon of each other.

Cauchy sequences are a very important concept in real analysis, and they are used to prove many important theorems, such as the Bolzano-Weierstrass theorem and the Cauchy-Schwarz inequality.

Completeness

A complete metric space is a metric space in which every Cauchy sequence converges to a limit. The real numbers are a complete metric space, but not all metric spaces are complete.

Completeness is a very important property for a metric space, and it has many important applications. For example, completeness is used to prove that the real numbers are a closed field, and it is also used to prove the existence of solutions to many important differential equations.

Metric Spaces

A metric space is a set X together with a distance function d: X × X → R that satisfies the following three properties:

1. d(x, y) ≥ 0 for all x, y ∈ X. 2. d(x, y) = 0 if and only if x = y. 3. d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X.

Metric spaces are a very important tool in real analysis, and they are used to define many important concepts, such as convergence, limits, and Cauchy sequences.

Continuity

A function f: X → Y is continuous at a point x₀ ∈ X if for any number epsilon > 0, there exists a number delta > 0 such that |f(x) - f(x₀)| 0| 0, we can find a number delta > 0 such that the output of the function is within epsilon of the output at the point x₀ for all inputs that are within delta of the input x₀.

Continuity is a very important property for a function, and it has many important applications. For example, continuity is used to prove that the real numbers are a continuous field, and it is also used to prove the existence of solutions to many important differential equations.

Derivatives

The derivative of a function f: X → Y at a point x₀ ∈ X is defined as

$$f'(x_0) = \lim_{h\to 0}\frac{f(x_0 + h) - f(x_0)}{h}.$$

In other words, the derivative of a function at a point is the limit of the difference quotient as the increment approaches zero.

Derivatives are a very important tool in real analysis, and they are used to find the slope of a function, to determine whether a function is increasing or decreasing, and to find the maximum and minimum values of a function.

Integrals

The integral of a function f: [a, b] → R is defined as

$$\int_a^b f(x) dx = \lim_{n\to\infty}\sum_{i=1}^n f(x_i) \Delta x,$$

where x_i = a + iΔ

Real Analysis via Sequences and Series (Undergraduate Texts in Mathematics)

by Bruce van Brunt

4.4 out of 5

Language	:	English
File size	:	5918 KB
Screen Reader	:	Supported
Print length	:	487 pages

FREE