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Real Analysis 1-4 (semesters 1-4)
    Basic notions. Axioms of the real numbers (semester 1, weeks 1-2; 0 problems)
        Fundaments of Logic (semester 1, week 1; 16 problems)
        Proving Techniques: Proof by Contradiction, Induction (semester 1, week 1; 20 problems)
            Fibonacci Numbers (semester 1, week 1; 6 problems)
        Solving Inequalities and Optimization Problems by Inequalities between Means (semester 1, week 1; 19 problems)
        Sets, Functions, Combinatorics (semester 1, week 2; 22 problems)
    Axioms of the real numbers (semester 1, weeks 2-4; 0 problems)
        Field Axioms (semester 1, week 2; 5 problems)
        Ordering Axioms (semester 1, week 2; 6 problems)
        The Archimedean Axiom (semester 1, week 2; 3 problems)
        Cantor Axiom (semester 1, week 2; 7 problems)
        The Real Line, Intervals (semester 1, week 3; 15 problems)
Problem 134

    How many (a) maxima (b) upper bounds of a set of real numbers can have?

    Difficulty: 2.


Problem 136

    Determine the minimum, maximum, infimum, supremum of the following sets (if they have any)!

     (1)  \(\displaystyle [1 , 2]\),      (2)  \(\displaystyle (1 , 2)\),      (3)  \(\displaystyle \{\frac1n: n \in \N^+ \}\),      (4)  \(\displaystyle \Q\),      (5)  \(\displaystyle \{\frac1n + {1 \over \sqrt{n}}: n \in \N^+ \}\),
     (6)  \(\displaystyle \{\root {n} \of 2: n \in \N^+ \}\),      (7)  \(\displaystyle \{x: x \in (0, 1) \cap \Q\}\),      (8)  \(\displaystyle \{\frac1n + {1 \over k}: n, k \in \N^+ \}\),
     (9)  \(\displaystyle \{\sqrt{n+1} - \sqrt{n}: n \in \N^+ \}\),      (10)  \(\displaystyle \{n + \frac1n: n \in \N^+\}\)

    Difficulty: 2.


Problem 137

    Are the following sets bounded from above or from below? What is the maximum, minmimum, supremum and infimum? Which set is convex?

    \(\displaystyle \emptyset \qquad \{1,2,3,\dots\} \qquad \{1,-1/2,1/3,-1/4,1/5,\dots\} \qquad \Q \qquad \R \qquad [1,2) \qquad (2,3] \qquad [1,2)\cup(2,3] \)

    Difficulty: 2.


Problem 138

    Let \(\displaystyle H\) be a subset of the reals. Which properties of \(\displaystyle H\) are expressed by the following formulas?
     (1)  \(\displaystyle (\forall x\in \R) (\exists y\in H) (x<y);\)
     (2)  \(\displaystyle (\forall x\in H) (\exists y\in \R) (x<y);\)
     (3)  \(\displaystyle (\forall x\in H) (\exists y\in H) (x<y).\)

    Difficulty: 2.


Problem 129

    Prove that \(\displaystyle \sqrt{2}\) is irrational.

    Difficulty: 3.


Problem 131

    Let \(\displaystyle a,b\in \Q\) and \(\displaystyle c,d\) be irrational. What can we say about the rationality of \(\displaystyle a+b\), \(\displaystyle a+c\), \(\displaystyle c+d\), \(\displaystyle ab\), \(\displaystyle ac\) and \(\displaystyle cd\)?

    Difficulty: 3.


Problem 132

    Prove that there is a rational and an irrational number in every open interval.

    Difficulty: 3.


Problem 139

    Let \(\displaystyle A\cap B\ne \emptyset\). What can we say about the connections among \(\displaystyle \sup A,\ \sup B\) and \(\displaystyle \sup (A\cup B) ,\ \sup(A\cap B)\) and \(\displaystyle \sup (A\setminus B)\)?

    Difficulty: 3.


Problem 141

    Which subsets \(\displaystyle H\subset\R\) satisfy that

       (a) \(\displaystyle \inf H<\sup H\);    (b) \(\displaystyle \inf H=\sup H\);    (c) \(\displaystyle \inf H>\sup H\)?

    Difficulty: 3.


Problem 130

    Prove that

     (1)  \(\displaystyle \sqrt 3\) is irrational;      (2)  \(\displaystyle {\sqrt 2 \over \sqrt 3}\) is irrational;      (3)  \(\displaystyle {{{{\sqrt 2 + 1} \over 2} + 3} \over 4} + 5\) is irrational!

    Difficulty: 4.


Problem 145

    Let \(\displaystyle a_n=\sqrt{n+1}+(-1)^n\sqrt{n}\).

    \(\displaystyle \inf\{a_n|n\in \N\}=?\)

    Difficulty: 4.


Problem 143

    What are the suprema and infima of the following sets?

    a) \(\displaystyle \{{1\over n}|n\in\N\}\).

    b) \(\displaystyle \{{1\over n}|n\in\N\}\cup\{0\}\).

    c) \(\displaystyle \{{1\over n}|n\in\N\}\cup\{{-1\over n}|n\in\N\}\).

    d) \(\displaystyle \{{1\over n^n}|n\in\N\}\cup \{2,3\}\).

    e) \(\displaystyle \{{\cos n\over n^n}|n\in\N\}\cup [-6,-5] \cup (100,101)\).

    Difficulty: 5.


Problem 144

    Let \(\displaystyle H,K\) be non-empty subsets of the real line \(\displaystyle \R\). What is the logical connection between the following two statements?

    a) \(\displaystyle \sup H < \inf K\);

    b) \(\displaystyle \forall x\in H\) \(\displaystyle \exists y\in K\) \(\displaystyle x<y\).

    Difficulty: 5.


Problem 146

    Let \(\displaystyle A,B\) be subsets of the real line \(\displaystyle \R\) such that \(\displaystyle A\cup B=(0,1)\). Does it imply that

    \(\displaystyle \inf A=0\qquad\hbox{or}\qquad \inf B=0\qquad?\)

    Difficulty: 5.


Problem 147

    Prove that all convex subset of \(\displaystyle \R\) are intervals.

    Difficulty: 7.


        Completeness Theorem, Connectivity, Topology of the Real Line. (semester 1, week 3; 4 problems)
        Powers (semester 1, week 4; 3 problems)
    Convergence of Sequences (semester 1, weeks 4-6; 0 problems)
        Theoretical Exercises (semester 1, week 4; 61 problems)
        Order of Sequences, Threshold Index (semester 1, week 4; 22 problems)
        Limit Points, liminf, limsup (semester 1, week 5; 15 problems)
        Calculating the Limit of Sequences (semester 1, week 5; 33 problems)
        Recursively Defined Sequences (semester 1, week 5; 21 problems)
        The Number $e$ (semester 1, week 5; 13 problems)
        Bolzano–Weierstrass Theorem and Cauchy Criterion (semester 1, week 6; 5 problems)
        Infinite Sums: Introduction (semester 1, week 6; 21 problems)
    Cardinalities of Sets (semester 1, week 7; 0 problems)
        Countable and not countable sets (semester 1, week 7; 6 problems)
        Not countable Sets (semester 1, week 7; 0 problems)
    Limit and Continuity of Real Functions (semester 1, weeks 7-10; 0 problems)
        Global Properties of Real Functions (semester 1, week 7; 20 problems)
        Continuity and Limits of Functions (semester 1, week 8; 33 problems)
        Calculating Limits of Functions (semester 1, week 8; 29 problems)
        Continuity and Convergent Sequences (semester 1, week 9; 0 problems)
        Continuous Functions on a Closed Bounded Interval (semester 1, week 9; 9 problems)
        Uniformly Continuous Functions (semester 1, week 10; 5 problems)
        Monotonity and Continuity (semester 1, week 10; 2 problems)
        Convexity and Continuity (semester 1, week 10; 7 problems)
    Elementary functions (semester 1, weeks 11-12; 0 problems)
        Arclength of the Graph of the Function (semester 1, week 11; 0 problems)
        Exponential, Logarithm, and Power Functions (semester 1, week 11; 17 problems)
        Inequalities (semester 1, week 12; 1 problems)
        Trigonometric Functions and their Inverses (semester 1, week 12; 3 problems)
    Differential Calculus and its Applications (semester 2, weeks 0-3; 0 problems)
        The Notion of Differentiation (semester 2, week 0; 47 problems)
        Tangents (semester 2, week 1; 11 problems)
        Higher Order Derivatives (semester 2, week 1; 13 problems)
        Local Properties and the Derivative (semester 2, week 1; 4 problems)
        Mean Value Theorems (semester 2, week 1; 3 problems)
        Number of Roots (semester 2, week 1; 5 problems)
        Exercises for Extremal Values (semester 2, week 2; 2 problems)
            Inequalities, Estimates (semester 2, week 2; 14 problems)
        The L'Hospital Rule (semester 2, week 2; 14 problems)
        Polynomial Approximation, Taylor Polynomial (semester 2, week 3; 20 problems)
        Convexity (semester 2, week 3; 5 problems)
        Analysis of Differentiable Functions (semester 2, week 3; 6 problems)
    Riemann Integral (semester 2, weeks 4-11; 0 problems)
        Definite Integral (semester 2, week 4; 11 problems)
        Indefinite Integral (semester 2, weeks 5-6; 13 problems)
        Properties of the Derivative (semester 2, week 6; 2 problems)
        Newton-Leibniz formula (semester 2, week 6; 2 problems)
        Integral Calculus (semester 2, week 7; 5 problems)
        Applications of the Integral Calculus (semester 2, week 8; 4 problems)
            Calculating the Area and the Volume (semester 2, week 8; 0 problems)
            Calculating the Arclength (semester 2, week 8; 3 problems)
            Surface Area of Surfaces of Revolution (semester 2, week 8; 0 problems)
        Integral and Inequalities (semester 2, week 8; 5 problems)
        Improper Integral (semester 2, week 9; 9 problems)
        Liouville Theorem (semester 2, week 10; 0 problems)
        Functions of Bounded Variation (semester 2, week 11; 2 problems)
        Riemann-Stieltjes integral (semester 2, week 11; 2 problems)
    Infinite Series (semester 2, weeks 12-13; 38 problems)
    Sequences and Series of Functions (semester 3, weeks 1-2; 0 problems)
        Convergence of Dequences of Functions (semester 3, week 1; 16 problems)
        Convergence of Series of Functions (semester 3, week 1; 17 problems)
        Taylor and Power Series (semester 3, week 2; 12 problems)
    Differentiability in Higher Dimensions (semester 3, weeks 3-6; 0 problems)
        Topology of the $n$-dimensional Space (semester 3, week 3; 29 problems)
        Real Valued Functions of Several Variables (semester 3, weeks 3-4; 0 problems)
            Limits and Continuity in $R^n$ (semester 3, week 3; 16 problems)
            Differentiation in $R^n$ (semester 3, week 4; 62 problems)
        Vector Valued Functions of Several Variables (semester 3, weeks 5-6; 0 problems)
            Limit and Continuity (semester 3, week 5; 3 problems)
            Differentiation (semester 3, week 5; 11 problems)
            Implicite functions (semester 3, week 6; 0 problems)
    Jordan Measure and Riemann Integral in Higher Dimensions (semester 3, weeks 7-9; 60 problems)
    Integral Theorems of Vector Calculus (semester 3, week 10 -- semester 4, week 1; 0 problems)
        The Line Integral (semester 3, week 10; 11 problems)
        Newton-Leibniz Formula (semester 3, week 11; 6 problems)
        Existence of the Primitive Function (semester 3, week 12; 13 problems)
        Integral Theorems in 2D (semester 4, week 1; 2 problems)
        Integral Theorems in 3D (semester 4, week 1; 12 problems)
    Measure Theory (semester 4, weeks 3-99; 0 problems)
        Set Algebras (semester 4, week 3; 9 problems)
        Measures and Outer Measures (semester 4, week 4; 8 problems)
        Measurable Functions. Integral (semester 4, week 5; 10 problems)
        Integrating Sequences and Series of Functions (semester 4, weeks 7-8; 11 problems)
        Fubini Theorem (semester 4, week 9; 1 problems)
        Differentiation (semester 4, weeks 11-12; 7 problems)
Complex Analysis (semester 5)
    Complex differentiability (semester 5, week 0; 0 problems)
        Complex numbers (semester 5, week 0; 21 problems)
            The Riemann sphere (semester 5, week 0; 1 problems)
    Regular functions (semester 5, weeks 1-2; 0 problems)
        Complex differentiability (semester 5, week 1; 7 problems)
        The Cauchy-Riemann equations (semester 5, week 1; 3 problems)
        Power series (semester 5, weeks 1-2; 0 problems)
            Domain of convergence (semester 5, week 2; 9 problems)
            Regularity of power series (semester 5, week 2; 2 problems)
            Taylor series (semester 5, week 2; 1 problems)
        Elementary functions (semester 5, week 2; 0 problems)
            The complex exponential and trigonometric functions (semester 5, week 2; 8 problems)
            Complex logarithm (semester 5, week 2; 12 problems)
    Complex Line Integral and Applications (semester 5, weeks 3-5; 0 problems)
        The complex line integral (semester 5, week 3; 9 problems)
        Cauchy's theorem (semester 5, week 3; 6 problems)
        The Cauchy formula (semester 5, week 4; 12 problems)
        Power and Laurent series expansions (semester 5, week 5; 0 problems)
            Power series expansion (semester 5, week 5; 2 problems)
            Liouville's Theorem (semester 5, week 5; 7 problems)
        Local properties of holomorphic functions (semester 5, week 5; 0 problems)
        Consequences of analyticity (semester 5, week 5; 8 problems)
            The maximum principle (semester 5, week 5; 4 problems)
        Laurent series (semester 5, week 5; 9 problems)
    Isolated singularities (semester 5, weeks 5-8; 0 problems)
        Singularities (semester 5, week 5; 4 problems)
        Cauchy's theorem on residues (semester 5, weeks 7-8; 15 problems)
            Residue calculus (semester 5, week 7; 6 problems)
            Applications (semester 5, week 7; 0 problems)
                Evaluation of series (semester 5, week 7; 7 problems)
                Evaluation of integrals (semester 5, week 7; 26 problems)
            The argument principle and Rouche's theorem (semester 5, week 8; 7 problems)
    Conformal maps (semester 5, weeks 9-10; 0 problems)
        Fractional linear transformations (semester 5, week 9; 20 problems)
        Riemann mapping theorem (semester 5, week 9; 11 problems)
        Schwarz lemma (semester 5, week 9; 12 problems)
        Caratheodory's theorem (semester 5, week 9; 2 problems)
        Schwarz reflection principle (semester 5, week 10; 2 problems)
    Harmonic functions (semester 5, weeks 11-12; 8 problems)
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