---
title: 18.100B problem set 01
description: attempts and solutions for 18.100B problem set 01
tags:
  - course
problems: {}
solutionTerm: '2006'
pageLayout: default
slug: courses/18-100b-analysis-i-fall-2010/psets/01
permalink: https://aarnphm.xyz/courses/18-100b-analysis-i-fall-2010/psets/01.md
generator:
  quartz: v4.6.0
  hostedProvider: Cloudflare
  baseUrl: aarnphm.xyz
full: https://aarnphm.xyz/llms-full.txt
---
Due Tuesday, September 14. Reading: Thu Sept.9: ordered sets and fields, Rudin 1.1-31.

## 1. Strict order from a partial order

> \[!question\] problem 1
>
> Suppose that $S$ is a set and $\preceq$ is a relation on $S$ with the following properties:
>
> - For all $x \in S$, $x \preceq x$.
> - For all $x, y \in S$, if $x \preceq y$ and $y \preceq x$ then $x = y$.
> - For all $x, y, z \in S$, if $x \preceq y$ and $y \preceq z$, then $x \preceq z$.
>
> Define a new relation $\prec$ on $S$ by $x \prec y$ iff $x \preceq y$ and $x \ne y$. Does this define an order conforming to Definition 1.5 in Rudin? If so, prove it; if not, exhibit a counterexample.

^problem-1

### attempt

## 2. Rudin Exercise 6, p. 22

> \[!question\] problem 2
>
> Exercise 6, p. 22 of Rudin. \[Here $b > 1$ is an element of $\mathbb{R}$ and you may use the ‘definition’ of $\mathbb{R}$ as ordered field with least upper bound property. Then recall that $y = x^{\frac{1}{n}}$ is defined as solution of $y^n = x$, $y \ge 0$.\]

^problem-2

### attempt

## 3. Lexicographic order on $\mathbb{C}$

> \[!question\] problem 3
>
> (Exercise 9 p. 22 of Rudin – lexicographic order) For complex numbers $z = a + bi \in \mathbb{C}$ and $w = c + di \in \mathbb{C}$ define "$z < w$" if either $a < c$ or if ($a = c$ and $b < d$). Prove that this turns $\mathbb{C}$ into an ordered set. Is this an ordered field? Does it have the least-upper-bound property?

^problem-3

### attempt

## 4. Archimedean property of $\mathbb{Q}$

> \[!question\] problem 4
>
> (a) Prove that the field $\mathbb{Q}$ of rational numbers has the Archimedean property.
>
> (b) Compare the least upper bound property with the Archimedean property – which one is ‘stronger’? Why?

^problem-4

### attempt

## 5. Proofs by induction

> \[!question\] problem 5
>
> Review the logic of a proof by induction. (The – not always reliable – wikipedia gives a good explanation in this case.)
>
> (a) Prove that $(1 + 2 + \cdots + n)^2 = 1^3 + 2^3 + \cdots + n^3$ for each $n \in \mathbb{N}$.
>
> (b) Find a proof of the Bernoulli inequality:
>
> $$
> (1 + x)^n \ge 1 + nx \qquad \text{for all } x \in \mathbb{R}, x \ge -1 \text{ and } n \in \mathbb{N}, n \ge 2.
> $$
>
> (not for credit) Show that strict inequality $(1 + x)^n > 1 + nx$ holds unless $x = 0$.

^problem-5

### attempt