Self-Study Notes (Understanding Analysis)

  1. Basics
    1. Prove that if \(p^2\) is even, then \(p\) is even.
    2. Prove that \(\sqrt{2}\) is irrational.
    3. Prove that \(\sqrt{3}\) is irrational.
    4. Two real numbers \(a\) and \(b\) are equal if and only if for every real number \(\epsilon > 0\) it follows that \(|a - b| < \epsilon\).
  2. The Absolute Value Function
    1. Definition and Properties
    2. Let \(x \in \mathbf{R}\), prove that \(|x| \geq 0\).
    3. Let \(x \in \mathbf{R}\), prove that \(-|x| \leq x \leq |x|\).
    4. For two real numbers \(x\) and \(y\), prove that \(\max(x,y) = \frac{1}{2}(x+y+|x - y|)\).
    5. \(|ab| = |a|\cdot|b|\) holds for all real numbers \(a\) and \(b\).
    6. \(|a+b| \leq |a|+|b|\) holds for all real numbers \(a\) and \(b\).
    7. \(|a - c| \leq |a - b|+|b - c|\) holds for all real numbers \(a\), \(b\) and \(c\).
  3. Upper and Lower Bounds
    1. Bounds Definitions
    2. [Lemma 1.3.8] If \(s\) is an upper bound for \(A\), then \(s = \sup A\) if and only if for every choice of \(\epsilon > 0\), there exists an element \(a \in A\) satisfying \(s - \epsilon < a\).
    3. The Nested Interval Property
    4. The Archimedean Principle
    5. [Density of \(\mathbf{Q}\) in \(\mathbf{R}\)] For every two real numbers \(a\) and \(b\) with \(a < b\), there exists a rational number \(r\) satisfying \(a < r < b\).

      6. Exercises

    6. Prove that if \(a\) is an upper bound for \(A\), and if \(a\) is also an element of \(A\), then it must be that \(a=\sup A\)
    7. Assume that \(A\) and \(B\) are nonempty, bounded above, and satisfy \(B \subseteq A\). Show \(\sup B \leq \sup A\).
    8. The set defined as \(c+A = \{c + a : a \in A\}\) where \(c\) is a constant has a least upper bound equal to \(c + sup A\).
    9. [1.3.5] The set defined as \(cA = \{ca : a \in A\}\) where \(c \geq 0\) has \(\sup cA = c\sup A\).
    10. [1.3.6] The set defined as \(A + B = \{a + b: a \in A \text{ and } b \in B\}\) has \(\sup (A+B) = \sup A + \sup B\).
    11. [1.3.6] The set defined as \(A + B = \{a + b: a \in A \text{ and } b \in B\}\) has \(\sup (A+B) = \sup A + \sup B\) (Alternative Proof).
    12. [1.3.4] Let \(A\) and \(B\) be nonempty and bounded above. Find a formula for \(\sup (A \cup B)\) and prove that it is correct.
    13. [1.3.11] If \(A\) and \(B\) are nonempty and bounded sets of real numbers such that \(A \subseteq B\) then \(\inf B \leq \inf A \leq \sup A \leq \sup B\).
    14. Let \(A = \{x \in \mathbf{R}: x < 0\}\). Then \(\sup(A) = 0\).
    15. Show that \(\inf\big(\{\frac{1}{n}: n \in \mathbf{N}\} \big) = 0.\)
    16. Show that \(\sup\big(\{\frac{1}{n}: n \in \mathbf{N}\} \big) = 1.\)
  4. Cardinality
    1. Definitions
    2. The open interval \((0,1) = \{x \in \mathbf{R}: 0 < x < 1\}\) is uncountable.
  5. Sequences and Subsequences
    1. Sequences Definitions
    2. Show the Limit Template + Example \(\lim\big(\frac{1}{\sqrt{n}}\big)= 0\).
    3. [Uniqueness of Limits (2.2.7)] The limit of a sequence, when it exists must be unique.
    4. [Bounded Sequences (2.3.1)] A sequence \(x_n\) is bounded if there exists a number \(M > 0\) such that every term in the sequence \(|x_n| \leq M\) for all \(n \in \mathbf{N}\).
    5. [Convergent Sequences (2.3.2)] Every convergent sequence is bounded.
    6. [The Algebraic Limit Theorem (2.3.3)] (i) \(\lim (ca_n) = ca\) for all \(c \in \mathbf{R}\).
    7. [The Algebraic Limit Theorem (2.3.3)] (ii) \(\lim (a_n + b_n) = a + b\).
    8. [The Algebraic Limit Theorem (2.3.3)] (iii) \(\lim (a_nb_n) = ab\).
    9. [The Order Limit Theorem ((2.3.4)]
    10. [Monotone Convergence Theorem (2.4.2)] If a sequence is monotone and bounded, then it converges.
    11. Subsequences Definitions
    12. [Convergence of Subsequence (2.5.2)] Subsequences of a convergent sequence converge to the same limit as the original sequence.
    13. [Bolzano-Weierstrass Theorem (2.5.5)] Every bounded sequence contains a convergent subsequence.
    14. (2.6.3) Cauchy Sequences are Bounded
    15. Every convergent sequence is a Cauchy sequence
    16. [Cauchy Criterion (2.6.4)] A sequence converges if and only if it is a Cauchy sequence.

      17. Exercises

    17. Show that \(\lim\big(\frac{n+1}{n}\big)= 1\).
    18. Let \(x_n \geq 0\) for all \(n \in \mathbf{N}\). Show that if \((x_n) \longrightarrow x\), Then \((\sqrt{x_n}) \longrightarrow \sqrt{x}\).
    19. [Squeeze Theorem] Show that if \(x_n \leq y_n \leq z_n\) for all \(n \in \mathbf{N}\), and if \(\lim x_n = \lim z_n = l\), then \(\lim y_n = l\) as well.
    20. Show that if \((|x_n|) \rightarrow 0\) for all \(n \in \mathbf{N}\), then \(x_n \rightarrow 0\).
    21. Show that \((\sqrt{n + 1} - \sqrt{n})\) converges to 0.
    22. Prove that \(b > b^2 > b^3 > b^4 > ... > 0\) converges to 0 if \(0 < b < 1\). (Example of 2.5.2).
    23. A sequence \(a_n\) converges to \(a\) if and only if every subsequence of \(a_n\) also converges to \(a\).
    24. If two subsequences of \(a_n\) converge to different limits, or if any subsequences of \(a_n\) diverges then \(a_n\) diverges.
    25. Prove that \((a_n) = (-1)^n\) diverges.
  6. Series
    1. Series Definitions
    2. (2.4.6) Cauchy Condensation Test
    3. (2.7.1) The Algebraic Limit Theorem for Series
    4. (2.7.2) Cauchy Criterion for Series
    5. (2.7.3) If the series \(\sum_{n=1}^{\infty} a_n\) converges then \((a_n) \rightarrow 0\).
    6. (2.7.4) Comparison Test for Series
    7. (2.7.5) The Geometric Series
    8. (2.7.6) [The Absolute Convergence Test] If the series \(\sum_{n=1}^{\infty}|a_n|\) converges, then \(\sum_{n=1}^{\infty} a_n\) converges as well.
    9. (2.7.7) The Alternating Series Test
  7. Topology of R
    1. Cantor Sets
    2. Open Sets
      1. (3.2.3) The union of an arbitrary collection of open sets is open and the intersection of a finite collection of open sets is open.
    3. Limit Points
    4. Closed Sets
      1. Prove that a set \(F \subseteq \mathbf{R}\) is closed if and only if it contains its limit points.
      2. Prove that a closed interval \([c,d] = \{c \leq x \leq d\}\) is a closed set.
    5. Closure
    6. (3.2.13) A set \(O\) is open if and only if \(O^c\) is closed. Likewise, a set \(F\) is closed if and only if \(F^c\) is open.
    7. Compact Sets
      1. (Theorem 3.3.4) [Characterization of Compactness in R] A set \(K \in \mathbf{R}\) is compact if and only if it is closed and bounded.
      2. (Theorem 3.3.5) [Nested Compact Set Property] If \( K_1 \supseteq K_2 \supseteq K_3 \supseteq K_4 \supseteq ...\) is a nested sequence of nonempty compact sets, then the intersection \(\bigcap_{n=1}^{\infty} K_n\) is not empty.
    8. Open Cover
      1. (Theorem 3.3.8) [Heine-Borel Theorem] Let \(K\) be a subset of \(\mathbf{R}\). All of the following statements are equivalent in the sense that any one of them implies the other two (i) \(K\) is compact. (ii) \(K\) is closed and bounded (iii) Every open cover for \(K\) has a finite subcover.
    9. Perfect Sets