# Exam 1 (Take Home Portion)

## Instructions

This is an open book, open notes take home exam that is due Wednesday, September 28, 2016.  This exam consists of six questions.  You must complete questions 1 through 4, but you may choose any two questions from the “additional exam questions” section below.  Please note that you may only choose two questions.  If you turn in work for more than two, I will only grade the first two problems I come across and ignore the others.

Proofs need to be written clearly and concisely to receive full credit (although accompanying pictures are encouraged, especially if they help clarify an idea or description).

While taking this exam you are allowed to use any of your notes from this course (including any of my blog posts on this website that are related to Analysis — and, yes, this includes homework solutions), and you are allowed to use Chapters 0 and 1 from our textbook.  Finally, you are also allowed to consult with me; I will try to be as helpful as I can when answering your questions.

As a last and unnecessary note: academic dishonesty of any kind will not be tolerated.  Such behavior will result in an automatic F for this class.  If you are not sure whether or not a resource you would like to use is in violation of our rules, then just ask — I’ll be more than happy to clarify.

## Mandatory Exam Questions (60 points)

Question 1.  (a) (15 points) Let $S$ be a set of real numbers that is bounded from below and let  $x = \inf S$.  Prove that $x \in S$ or that $x$ is an accumulation point for $S$.

(b) (5 points) Provide an example of a set $S \subseteq \mathbb{R}$ that is bounded below, where $x = \inf S$ is an element of $S$, but where $x$ is not an accumulation point for $S$ (or explain why no such example exists).

Question 2.  (10 points) Prove that the sequence  of real numbers $\{a_n\}$ where

$\displaystyle a_n = \frac{5n}{3n+2}$

converges.

Question 3.  (20 points) For this problem we will need a new definition.

Def. (Bounded subset of a metric space)  Suppose $(X, d)$ is a metric space.  Then a subset $S \subseteq X$ is said to be bounded if there exists a non-negative real number $R \geq 0$ and if there exists a point $x_0 \in X$ so that

$\displaystyle S \subseteq B_R\left(x_0\right)$.

(In more colloquial terms: a subset $S \subseteq X$ is said to be bounded if it can be contained in a finite-radius ball.  Recall from our homework that a(n open) ball in a metric space is defined as $B_R(x_0) = \{ x \in X : d(x, x_0) < R \}$.)

Parts (a)-(c) of this question concern the metric space $(X, d)$ where $X = \mathbb{R}$ and $d$ is given by

$\displaystyle d(x,y) = 0 \text{ if } x = y$ and $\displaystyle d(x,y) = 1 \text{ if } x \neq y.$

(a) (4 points) Prove or disprove: the entire set $X = \mathbb{R}$ is bounded.

(b) (8 points) Suppose $\{a_n\}$ is a convergent sequence of points in this metric space.  What can you say about it?  Phrase your answer as a theorem (one that is of the form “If $\{a_n\}$ is a convergent sequence of points in this metric space, then…”) and provide a proof.

(c ) (8 points)  Is the Bolzano-Weierstrass Theorem true in this metric space?  Prove your answer.

Question 4. (10 points) Describe an idea you had for a problem (or set of problems) from this class that did not work, and discuss what you learned from this experience.

## Additional Exam Questions (40 points or more)

Question 5. (25 points)   Consider a sequence of real numbers $\displaystyle \left\{a_n\right\}$ that is bounded.  The Completeness Axiom for $\mathbb{R}$ tells us that the supremum of this sequence exists; say

$\displaystyle s_1 = \sup \left\{a_1, a_2, a_3, a_4, a_5, \cdots \right\}.$

Define another sequence of real numbers $\displaystyle \left\{s_n\right\} = \left\{ s_1, s_2, s_3, \cdots, s_n, \cdots \right \}$ where

$\displaystyle s_1 = \sup \left\{a_1, a_2, a_3, a_4, a_5, \cdots \right\}$

$\displaystyle s_2 = \sup \left\{a_2, a_3, a_4, a_5, \cdots \right\}$

$\displaystyle s_3 = \sup \left\{a_3, a_4, a_5, \cdots \right\}$

and, more generally,

$\displaystyle s_n = \sup \left\{a_n, a_{n+1}, a_{n+2}, \cdots \right\}$.

If this sequence of suprema converges, we call its limit ithe eventual supremum of the original sequence $\{a_n\}$, and we write

$\displaystyle \lim_{n\to\infty} s_n = \text{e-sup} \left\{a_n\right\}.$

(a) (10 points) Prove that the sequence of suprema, $\displaystyle \left\{ s_n \right\}$, always converges, i.e. that the eventual supremum always exists, whenever $\{a_n\}$ is bounded.  (Note: this happens whether or not the original sequence $\{a_n\}$ converges.)

(b) (15 points) Prove that a sequence of real numbers $\{a_n\}$ converges to $a$ if and only if

$\displaystyle \text{e-inf} \left\{a_n\right\} = a = \text{e-sup} \left\{a_n\right\}$

where the eventual infimum, $\text{e-inf} \{a_n\}$, is defined in a way similar to that of $\text{e-sup}\{a_n\}$.

Question 6.  (a) (10 points) Let $\{a_n\}$ be a sequence of real numbers with an accumulation point $a \in \mathbb{R}$.  Prove or disprove: $\{a_n\}$ has a subsequence that converges to $a$.

(b) (10 points) Let $\{a_n\}$ be a bounded sequence of real numbers.  Prove or disprove: $\{a_n\}$ has a convergent subsequence.

Question 7.  (20 points) In class we discussed The Nested Interval Property of $\mathbb{R}$.  Prove this slightly different version:

Theorem.  Suppose

$\cdots \subset I_n \subset \cdots \subset I_3 \subset I_2 \subset I_1$

is a nested sequence of closed, non-empty, bounded intervals of real numbers.  Let $\ell_n$ denote the length of the interval $I_n$, and suppose that the sequence of these lengths converges to zero, i.e. that

$\displaystyle \lim_{n\to\infty} \ell_n = 0.$

Prove that the intersection $\displaystyle \bigcap_{n=1}^{\infty} I_n$ contains exactly one real number.

Question 8.  (20 points) Consider the sequence of real numbers $\{a_n\}$ where

$\displaystyle a_n = \sqrt{n}$.

Prove that the sequence $\{b_n\}$ converges to $0$ where

$\displaystyle b_n = \left|a_{n+1}-a_n\right| = a_{n+1}-a_n$.

Also prove that, however, the original sequence $\{a_n\}$ is not Cauchy.

Question 9.  (20 points) Suppose $\{a_n\}$ is a sequence of positive real numbers and that the sequence $\{b_n\}$ converges to $b < 1$ where

$\displaystyle b_n = \frac{a_{n+1}}{a_n}$.

Prove that $\{a_n\}$ converges to $0$.

Question 10. (22 points)  A sequence of real numbers $\{a_n\}$ is said to be a contractive sequence if there exists a real number $c \in (0, 1)$ where, for every $k \in \mathbb{N}$, it follows that

$\displaystyle \left|a_{k+1} - a_k\right| \leq c\left|a_k - a_{k-1}\right|.$

(a) (5 points) Prove that if $\{a_n\}$ is a contractive sequence with constant $c \in (0, 1)$, then for every $n \in \mathbb{N}$ it follows that

$\displaystyle \left|a_{n+1}-a_n\right| \leq c^{n-1}\left|a_2 - a_1\right|.$

(b) (2 points) Define the notion of a contractive sequence for a sequence of points in an arbitrary metric space $(X, d)$.

(c) (15 points) Prove or disprove: every contractive sequence of real numbers converges.