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Feb 5, 2010 · SECTION 1.3 introduces basic ideas of set theory in the context of sets of

**real**num-bers. In this section we prove two fundamental theorems: the Heine–Borel and Bolzano– Weierstrass theorems. 1.1 THE**REAL**NUMBER SYSTEM Having taken calculus, you know a lot about the**real**number system; however, you prob-Abstract. These are some notes on introductory

**real****analysis**. They cover the properties of the**real**numbers, sequences and series of**real**numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration. They don’t include multi-variable calculus or contain any problem sets.ISBN: 9781718862401. [JL] = Basic

**Analysis**: Introduction to**Real****Analysis**(**Vol**. 1) (PDF - 2.2MB) by Jiří Lebl, June 2021 (used with permission) This book is available as a free PDF download. You can purchase a paper copy by following a link at the same site.From here, there are some very important deﬁnitions in

**real****analysis**. We say that b 0 is the least upper bound,orthesupremumofEif A) b 0 isanupperboundforEand B) ifbisanupperboundforEthenb 0 b: Wedenotethisasb 0 = supE. Similarly,wesaythatc 0 isthegreatestlowerbound,ortheinﬁnimumofEif A) c 0 isalowerboundforEand B) ifcisalowerboundforEthenc ...The foundations of

**real****analysis**are given by set theory, and the notion of cardinality in set theory, as well as the axiom of choice, occur frequently in analysis.in the course lectures for MATH 324 and 325 (

**Real****Analysis**I, II). You will ﬁnd that the lectures and these notes are very closely aligned. The notes highlight the important ideas and examples that you shouldIntroduction to

**Real****Analysis**Joshua Wilde, revised by Isabel ecu,T akTeshi Suzuki and María José Boccardi August 13, 2013 1 Sets Sets are the basic objects of mathematics. In fact, they are so basic that there is no simple and precise de nition of what a set actually is. orF our purposes it su ces to think of a set as a collection of objects.also include some additional material on

**real**powers and the exponential function. I recommend you refer to these notes for learning the mathematical content of the course, and refer to the textbook for examples, pictures, and additional exercises.**Real****Analysis****is****all****about****formalizing****and**making precise, a good deal of the intuition that resulted in the basic results in Calculus. As it turns out, the intuition is spot on, in several instances, but in some cases (and this is really why Real Analysis is important atan integrated manner, the core areas of

**analysis**. The objective was to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of**analysis**to other ﬂelds of mathematics and science. The present series of books is an elaboration of the lectures that were given.