Real Analysis

Self-Study Notes (Understanding Analysis)

1. 1.2 Preliminaries
   • Definition and Properties

   Exercises

   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 \(|a - b| < \epsilon\).
   5. Definition and Properties
   6. Let \(x \in \mathbf{R}\), prove that \(|x| \geq 0\).
   7. Let \(x \in \mathbf{R}\), prove that \(-|x| \leq x \leq |x|\).
   8. For two real numbers \(x\) and \(y\), prove that \(\max(x,y) = \frac{1}{2}(x+y+|x - y|)\).
   9. \(|ab| = |a|\cdot|b|\) holds for all real numbers \(a\) and \(b\).
   10. \(|a+b| \leq |a|+|b|\) holds for all real numbers \(a\) and \(b\).
   11. \(|a - c| \leq |a - b|+|b - c|\) holds for all real numbers \(a\), \(b\) and \(c\).

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2. 1.3 Axiom of Completeness (Upper and Lower Bounds)
   • Bounds Definitions
   • [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 \(s - \epsilon < a\).

   Exercises

   1. Compute the suprema and infima of the following sets ...
   2. Prove that if \(a\) is an upper bound for \(A\), and if \(a\) is also an element of \(A\), then it must be that \(a=\sup A\)
   3. Assume that \(A\) and \(B\) are nonempty, bounded above, and satisfy \(B \subseteq A\). Show \(\sup B \leq \sup A\).
   4. The set defined as \(c+A = \{c + a : a \in A\}\) where \(c\) is a constant has a least upper bound equal to \(c + sup A\).
   5. [1.3.5] The set defined as \(cA = \{ca : a \in A\}\) where \(c \geq 0\) has \(\sup cA = c\sup A\).
   6. [1.3.6] The set defined as \(A + B = \{a + b: a \in A \text{ and } b \in B\}\) has \(\sup (A+B) = \sup A + \sup B\).
   7. [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).
   8. [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.
   9. [1.3.11] If \(A\) and \(B\) are nonempty and bounded sets of real numbers such that \(A \subseteq B\) then \(\inf B \leq \inf A \leq \sup A \leq \sup B\).
   10. Let \(A = \{x \in \mathbf{R}: x < 0\}\). Then \(\sup(A) = 0\).
   11. If \(\sup A < \sup B\), then show that there exists an element \(b \in B\) that is an upper bound for \(A\).
   12. Determine if the following statements are true or false.
   13. Let \(A\) and \(B\) be nonempty subsets of \(\mathbb{R}\) with \(A\) bounded above and \(B\) bounded below. Define \(A - B = \{a - b: a \in A, b \in B\}\). Prove that \(\sup(A - B) = \sup A - \inf B\).
   14. Prove that for any nonempty set \(S\) of real numbers that is bounded below, \(\inf S = -\sup(-S)\) where \(-S = \{-s: s \in S\}\).
   15. Let \(A\) be a nonempty subset of \(\mathbb{R}\) that is bounded above. Define \(S = \{ x \) is an upper bound for \(A\}\). Prove that \(\inf S = \sup A\).

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3. 1.4 Consequences of Completeness
   • The Nested Interval Property
   • The Archimedean Principle
   • [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 \(a < r < b\).

   Exercises

   1. Show that \(\inf\big(\{\frac{1}{n}: n \in \mathbf{N}\} \big) = 0.\)
   2. Show that \(\sup\big(\{\frac{1}{n}: n \in \mathbf{N}\} \big) = 1.\)

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4. 1.5 Cardinality
   • Definitions
   • The open interval \((0,1) = \{x \in \mathbf{R}: 0 < x < 1\}\) is uncountable.

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5. 2.2 The Limit of a Sequence
   • Sequences Definitions
   • [Uniqueness of Limits (2.2.7)] The limit of a sequence, when it exists must be unique.

   Exercises

   1. \(\lim\big(\frac{1}{\sqrt{n}}\big)= 0\). (Show the Limit Template)
   2. Prove that \(\lim\big(\frac{n+1}{n}\big)= 1\).
   3. Prove that \(\lim\big(\frac{1}{3n^3}\big)= 0\).
   4. Prove that \(\lim\big(n\big)\) diverges.
   5. Let \(x_n \geq 0\) for all \(n \in \mathbf{N}\). Show that if \((x_n) \longrightarrow x\), Then \((\sqrt{x_n}) \longrightarrow \sqrt{x}\).
   6. Let \(\{a_n\}\) be a sequence. Prove that if \(\lim\limits_{n \rightarrow \infty} a_n = L\) and \(\lim\limits_{n \rightarrow \infty} a_{n+1} = M\), then \(L = M\).
   7. If \(\{a_n\}\) is a convergent sequence with \(\lim\limits_{n \rightarrow \infty} a_n = L\). Prove that \(\{ |a_n|\}\) converges as well. Does \(\{ |a_n|\}\) converges to \(L\)?
   8. Verify using the definition of convergence that the following sequences converge to the proposed limit.
   9. Consider the sequence \(\{a_n\}\) defined by \(a_n = \frac{(-1)^nn}{n+1}\). Determine whether this sequence converges or diverges. If it converges, find its limit.
   10. Let \(\{a_n\}\) be a sequence. Prove that if \(\lim\limits_{n \rightarrow \infty} |a_n| = 0\), then \(\lim\limits_{n \rightarrow \infty} a_n = 0\).

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6. 2.3 The Algebraic and Order Limit Theorems
   • [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 \(n \in \mathbf{N}\).
   • [Convergent Sequences (2.3.2)] Every convergent sequence is bounded.
   • [The Algebraic Limit Theorem (2.3.3)] (i) \(\lim (ca_n) = ca\) for all \(c \in \mathbf{R}\).
   • [The Algebraic Limit Theorem (2.3.3)] (ii) \(\lim (a_n + b_n) = a + b\).
   • [The Algebraic Limit Theorem (2.3.3)] (iii) \(\lim (a_nb_n) = ab\).
   • [The Order Limit Theorem ((2.3.4)]
   • [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.

   Exercises

   1. Show that if \((|x_n|) \rightarrow 0\) for all \(n \in \mathbf{N}\), then \(x_n \rightarrow 0\).
   2. Argue that the sequence \(1,0,1,0,0,1,0,0,0,1,0,0,0,0,1,...\) does not converge to zero. For what values of \(\epsilon > 0\) does there exist a response \(N\)? For which values of \(\epsilon > 0\) is there no suitable response.
   3. Informally speaking, the sequence \(\sqrt{n}\) "converges to infinity". (a) Create a rigorous definition for the statement \(\lim\limits_{n \rightarrow \infty} x_n = \infty\) ...
   4. Create a rigorous definition for the statement \(\lim\limits_{n \rightarrow \infty} x_n = -\infty\). Give an example of a sequence that satisfies this definition.
   5. Construct a sequence \(\{a_n\}\) such that \(a_n > 0\) for all \(n \in \mathbb{N}\), \(\lim\limits_{n \rightarrow \infty} a_n = 0\), but \(\lim\limits_{n \rightarrow \infty} \frac{1}{a_n} = \infty\).
   6. Prove or disprove: If \(\lim\limits_{n \rightarrow \infty} a_n = 0\) and \(\{b_n\}\) is a bounded sequence, then \(\lim\limits_{n \rightarrow \infty} a_nb_n = 0\).

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7. 2.4 The Monotone Convergence Theorem
   • [Monotone Convergence Theorem (2.4.2)] If a sequence is monotone and bounded, then it converges.

   Exercises

   1. Let \(c\) be a real number with \(|c| < 1\). Using the monotone convergence theorem show that \(c^n \rightarrow 0\).
   2. Use the monotone convergence theorem to show that the following sequence is convergent: \(a_n = 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{n!}\)

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8. Subsequences
   • Subsequences Definitions
   • [Convergence of Subsequence (2.5.2)] Subsequences of a convergent sequence converge to the same limit as the original sequence.
   • [Bolzano-Weierstrass Theorem (2.5.5)] Every bounded sequence contains a convergent subsequence.
   • (2.6.3) Cauchy Sequences are Bounded
   • Every convergent sequence is a Cauchy sequence
   • [Cauchy Criterion (2.6.4)] A sequence converges if and only if it is a Cauchy sequence.

   Exercises

   1. If two subsequences of \(a_n\) converge to different limits, or if any subsequences of \(a_n\) diverges then \(a_n\) diverges.
   2. Prove that \((a_n) = (-1)^n\) diverges.
   3. A sequence \(a_n\) converges to \(a\) if and only if every subsequence of \(a_n\) also converges to \(a\).
   4. Prove that \(b > b^2 > b^3 > b^4 > ... > 0\) converges to 0 if \(0 < b < 1\). (Example of 2.5.2).

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9. Series
   • Series Definitions
   • (2.4.6) Cauchy Condensation Test
   • (2.7.1) The Algebraic Limit Theorem for Series
   • (2.7.2) Cauchy Criterion for Series
   • (2.7.3) If the series \(\sum_{n=1}^{\infty} a_n\) converges then \((a_n) \rightarrow 0\).
   • (2.7.4) Comparison Test for Series
   • (2.7.5) The Geometric Series
   • (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.
   • (2.7.7) The Alternating Series Test

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10. 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