Real Analysis and Multivariable Calculus: Graduate Level
                    Problems and Solutions
                        Igor Yanovsky
                           1
                  Real Analysis and Multivariable Calculus              Igor Yanovsky, 2005         2
                  Disclaimer:    This handbook is intended to assist graduate students with qualifying
                  examination preparation. Please be aware, however, that the handbook might contain,
                  and almost certainly contains, typos as well as incorrect or inaccurate solutions. I can
                  not be made responsible for any inaccuracies contained in this handbook.
                     Real Analysis and Multivariable Calculus                       Igor Yanovsky, 2005              3
                     Contents
                     1 Countability                                                                                  5
                     2 Unions, Intersections, and Topology of Sets                                                   7
                     3 Sequences and Series                                                                          9
                     4 Notes                                                                                       13
                         4.1   Least Upper Bound Property . . . . . . . . . . . . . . . . . . . . . . . .           13
                     5 Completeness                                                                                14
                     6 Compactness                                                                                 16
                     7 Continuity                                                                                  17
                         7.1   Continuity and Compactness . . . . . . . . . . . . . . . . . . . . . . . .           18
                     8 Sequences and Series of Functions                                                           19
                         8.1   Pointwise and Uniform Convergence . . . . . . . . . . . . . . . . . . . .            19
                         8.2   Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . .         19
                         8.3   Equicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       21
                               8.3.1   Arzela-Ascoli Theorem . . . . . . . . . . . . . . . . . . . . . . . .        21
                     9 Connectedness                                                                               21
                         9.1   Relative Topology      . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   21
                         9.2   Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        21
                         9.3   Path Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         23
                     10 Baire Category Theorem                                                                     24
                     11 Integration                                                                                26
                         11.1 Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        26
                         11.2 Existence of Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . .         27
                         11.3 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . .             27
                     12 Differentiation                                                                             30
                         12.1 R → R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       30
                               12.1.1 The Derivative of a Real Function . . . . . . . . . . . . . . . . .           30
                               12.1.2 Rolle’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .         30
                               12.1.3 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . .            30
                         12.2 R → Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        31
                         12.3 Rn → Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         31
                               12.3.1 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        34
                               12.3.2 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . .            35
                               12.3.3   ∂ (∂f) = ∂ (∂f) . . . . . . . . . . . . . . . . . . . . . . . . . . .       36
                                       ∂x ∂y       ∂y ∂x
                         12.4 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        37
                         12.5 Lagrange Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        40
                     Real Analysis and Multivariable Calculus                        Igor Yanovsky, 2005              4
                     13 Successive Approximations and Implicit Functions                                            41
                         13.1 Contraction Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . .           41
                         13.2 Inverse Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .           41
                         13.3 Implicit Function Theorem         . . . . . . . . . . . . . . . . . . . . . . . . .    44
                         13.4 Differentiation Under Integral Sign . . . . . . . . . . . . . . . . . . . . .           46