Press "Enter" to skip to content

Basic analysis: Introduction to real analysis by Jiri Lebl

By Jiri Lebl

A primary direction in mathematical research. Covers the true quantity method, sequences and sequence, non-stop services, the by-product, the Riemann imperative, sequences of services, and metric areas. initially constructed to coach Math 444 at collage of Illinois at Urbana-Champaign and later superior for Math 521 at collage of Wisconsin-Madison. See

Show description

Read or Download Basic analysis: Introduction to real analysis PDF

Similar introductory & beginning books

Beginning Game Programming

This ebook offers an advent to the whole box of video game programming. As readers paintings during the publication, they'll produce operating video games: one in second and one in 3D--offering a great creation to DirectX programming. starting with an advent to uncomplicated home windows programming, this publication quick advances to the fundamentals of DirectX programming, relocating up from surfaces to textures after which to 3D types.

Basic Teachings of the Great Philosophers

A whole precis of the perspectives of an important philosophers in Western civilization. each one significant box of philosophic inquiry includes a separate bankruptcy for better accessibility. contains Plato, Descartes, Spinoza, Kant, Hegel, Dewey, Sartre, and so on.

How to Make a Quilt: Learn Basic Sewing Techniques for Creating Patchwork Quilts and Projects. A Storey BASICS® Title

With basic step by step directions that require basically uncomplicated stitching abilities, Barbara Weiland Talbert exhibits you ways to make your personal attractive and sturdy quilts. Taking you thru the entire quilting strategy in an easy-to-follow series, Talbert exhibits you ways to choose an appropriate layout, decide upon the easiest cloth, reduce shapes, piece jointly blocks, gather the cover best, and end your venture.

Additional resources for Basic analysis: Introduction to real analysis

Example text

In fact, the cardinality of R is the same as the cardinality of P(N), although we will not prove this claim. 1 (Cantor). R is uncountable. 36 CHAPTER 1. REAL NUMBERS We give a modified version of Cantor’s original proof from 1874 as this proof requires the least setup. Normally this proof is stated as a contradiction proof, but a proof by contrapositive is always easier to understand. Proof. Let X ⊂ R be a countable subset such that for any two numbers a < b, there is an x ∈ X such that a < x < b.

That means that there exists a B such that xn ≤ B for all n, that is the set {xn | n ∈ N} is bounded. Let x := sup{xn | n ∈ N}. Let ε > 0 be arbitrary. As x is the supremum, then there must be at least one n0 ∈ N such that xn0 > x − ε (because x is the supremum). As {xn } is monotone increasing, then it is easy to see (by induction) that xn ≥ xn0 for all n ≥ n0 . Hence |xn − x| = x − xn ≤ x − xn0 < ε. 1. SEQUENCES AND LIMITS 43 Hence the sequence converges to x. We already know that a convergent sequence is bounded, which completes the other direction of the implication.

Let x := lim xn and y := lim yn . Let z := xy. Let ε > 0 be given. As {xn } is convergent, it is bounded. Therefore, find a B > 0 such that ε |xn | ≤ B for all n ∈ N. Find an M1 such that for all n ≥ M1 we have |xn − x| < 2|y| . Find an M2 such ε . Take M := max{M1 , M2 }. For all n ≥ M we have that for all n ≥ M2 we have |yn − y| < 2B |zn − z| = |(xn yn ) − (xy)| = |xn yn − (x + xn − xn )y| = |xn (yn − y) + (xn − x)y| ≤ |xn (yn − y)| + |(xn − x)y| = |xn | |yn − y| + |xn − x| |y| ≤ B |yn − y| + |xn − x| |y| ε ε |y| = ε.

Download PDF sample

Rated 4.32 of 5 – based on 24 votes