assignments

Assignment 1

• If \( \sup A < \sup B \) then show that there exists an element \( b \in B \) that is an upper bound for \( A\). (1.3.8 in 1st ed.)
• Do one of the following two (1.4.4 and 1.4.5 of 1st ed.):
  • Use the Archimedean property of \( \mathbb{R} \) to rigorously prove that \( \inf \{ 1/n : n \in \mathbb{N} \} = 0 \)
  • Prove that \( \cap_{n=1}^{\infty} (0, 1/n) = \emptyset \).
• Show that the set of all finite subsets of \( \mathbb{N} \) is a countable set. (1.4.10 in 1st ed.)
• Do at least three more of your own choosing from Chapter 1, including at least one from the self-guided section (1.5 or 1.6 in 1st or 2ns edition respectively).

Assignment 2

• Assume that \( (a_n) \) is a bounded sequence with the property that every convergent subsequence of \( (a_n) \) converges to the same limit \( a \) of \( \mathbb{R} \). Show that \( (a_n) \) must converge to \( a \). (2.5.4 in 1st ed.)
• Show that if \( x_n \leq y_n \leq z_n \) for all \( n \in \mathbb{N} \), and if \( \lim x_n = \lim z_n = l \), then \( \lim y_n = l \) as well. (2.3.3 in 1st ed.)
• Give an example of each of the following, or argue that such a request is impossible:
  • A Cauchy sequence that is not monotone.
  • A monotone sequence that is not Cauchy.
  • A Cauchy sequence with a divergent subsequence.
  • An unbounded sequence containing a subsequence that is Cauchy.
• Do at least three more of your own choosing from Chapter 2, including at least one from the self-guided section (2.8) and at least one that involves a proof starting "Let \( \epsilon > 0 \)".

Quiz 1

Assignment 3

• (3.2.14 in 1st Ed.) A set \( A \) is called an \( F_{\sigma} \) set if it can be written as the countable union of closed sets. A set \( B \) is called a \(G_{\delta} \) set if it can be written as the countable intersection of open sets.
  • Show that a closed interval \( [ a,b ] \) is a \( G_{\delta} \) set.
  • Show that the half-open interval \( (a,b] \) is both a \( F_{\sigma} \) and \( G_{\delta} \) set.
  • Show that \( \mathbb{Q} \) is an \( F_{\sigma} \) set and the set of irrationals \( \mathbb{I} \) forms a \( G_{\delta} \) set.
• (4.3.7 in 1st Ed.) Assume \( h : \mathbb{R} \rightarrow \mathbb{R} \) is continuous on \( \mathbb{R} \) and let \( K = \{x : h(x)=0 \} \). Show that \( K \) is a closed set.
• (4.5.7 in 1st Ed.) Let \( f \) be a continuous function on the closed interval \( [0,1] \) whose range is contained in \( [0,1] \). Prove that \( f \) must have a fixed point; that is, show \( f(x) = x \) for at least one value of \( x \in [0,1] \).
  • Addition from Matt: What happens if we loosen the domain and possible range to the open interval \( (0,1 ) \)? Is the result now false?
• Do at least three more of your own choosing from Chapters 3 and 4, including at least one from the self-guided section (4.6).

Assignment 4

• (5.2.3 in 1st Ed.) By imitating the Dirichlet constructions in Section 4.1, construct a function on \( mathbb{R} \) that is differentiable at a single point.
• (5.3.5 in 1st Ed.) A fixed point of a function \( f\) is a value \( x \) where \( f(x) = x \). Show that if \( f \) is differentiable on an interval with \( f'(x) \neq 1 \), then \( f \) can have at most one fixed point.
• (5.3.10 in 1st Ed.) Let \( f \) be a bounded function and assume \( \lim_{x \rightarrow c} g(x) = \infty \). Show that \( \lim_{x \rightarrow c} f(x)/g(x) = 0 \).
• Do at least three more of your own choosing from Chapters 5.

Quiz 2

Assignment 5

• (6.3.1, 1st ed)
  • (a) Let \( h_n (x) = \sin(nx)/n \). Show that \(h_n \rightarrow 0\) uniformly on \(\mathbb{R}\). At what points does the sequence of derivatives \(h_n'\) converge?
  • (b) Modify this example to show that it is possible for a sequence \( (f_n) \) to converge uniformly but for \( (f_n') \) to be unbounded.
• (6.6.2, 1st ed) Starting from the identity in equation (1) [for \( 1/(1-t)\)] of this section, find a power series representation for \( \ln(1+x) \). For which values of \(x\) is this expression valid? Added question: repeat the question, but this time use the formula for power series rather than (1).
• Do at least three more of your own choosing from Chapter 6.

Quiz 3

Assignment 6

• (7.2.6 in first edition.) Let \( f : [a,b] \rightarrow \mathbb{R} \) be increasing on the set \( [a,b] \). Show that \( f \) is integrable on \( [a,b] \).
• (7.5.7 in first edition.) If \( g \) is continuous on \( [a,b] \), show that there exists a point \( c \in (a,b) \) where \( g(c) = \frac{1}{b-a} \int_a^b g \).
• At least three more of your own choosing from Chapter 7, including one to do with the Fundamental Theorem and one to do with Lebesgue's criterion or measure.

Assignment 7

In-class Questions

Final Exam

Overall Grade