Lectures on Real Analysis (Australian Mathematical Society Lecture Series, Series Number 21)

This Is A Rigorous Introduction To Real Analysis For Undergraduate Students, Starting From The Axioms For A Complete Ordered Field And A Little Set Theory. The Book Avoids Any Preconceptions About The Real Numbers And Takes Them To Be Nothing But The Elements Of A Complete Ordered Field. All Of The Standard Topics Are Included, As Well As A Proper Treatment Of The Trigonometric Functions, Which Many Authors Take For Granted. The Final Chapters Of The Book Provide A Gentle, Example-based Introduction To Metric Spaces With An Application To Differential Equations On The Real Line. The Author's Exposition Is Concise And To The Point, Helping Students Focus On The Essentials. Over 200 Exercises Of Varying Difficulty Are Included, Many Of Them Adding To The Theory In The Text. The Book Is Perfect For Second-year Undergraduates And For More Advanced Students Who Need A Foundation In Real Analysis-- Machine Generated Contents Note: Preface; To The Student; 1. Numbers, Sets, And Functions; 2. The Real Numbers; 3. Sequences; 4. Open, Closed, And Compact Sets; 5. Continuity; 6. Differentiation; 7. Integration; 8. Sequences And Series Of Functions; 9. Metric Spaces; 10. The Contraction Principle; Index. Finnur Lárusson, University Of Adelaide. Includes Index. "This is a rigorous introduction to real analysis for undergraduate students, starting from the axioms for a complete ordered field and a little set theory. The book avoids any preconceptions about the real numbers and takes them to be nothing but the elements of a complete ordered field. All of the standard topics are included, as well as a proper treatment of the trigonometric functions, which many authors take for granted. The final chapters of the book provide a gentle, example-based introduction to metric spaces with an application to differential equations on the real line. The author's exposition is concise and to the point, helping students focus on the essentials. Over 200 exercises of varying difficulty are included, many of them adding to the theory in the text. The book is perfect for second-year undergraduates and for more advanced students who need a foundation in real analysis"-- Provided by publisher 00. 9781107608528 01.0_pp_i_iv_Frontmatter 02.0_pp_v_vi_Contents 03.0_pp_vii_viii_Preface 04.0_pp_ix_x_To_the_student 05.0_pp_1_14_Numbers_sets_and_functions 06.0_pp_15_22_The_real_numbers 07.0_pp_23_38_Sequences 08.0_pp_39_44_Open_closed_and_compact_sets 09.0_pp_45_54_Continuity 10.0_pp_55_62_Difierentiation 11.0_pp_63_72_Integration 12.0_pp_73_90_Sequences_and_series_of_functions 13.0_pp_91_102_Metric_spaces 14.0_pp_103_112_The_contraction_principle 15.0_pp_113_117_Index