\documentclass{article}
\usepackage{amssymb}
\usepackage{multicol}
\usepackage{graphicx}
\usepackage{color}
\usepackage{amsthm}
\usepackage{hyperref}
\usepackage{amsmath}
\usepackage{verbatim}
\usepackage{caption}
\usepackage{subcaption}
\usepackage{amsmath, epsfig}
\setlength{\parskip}{8pt}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{corollary}{Corollary}
%\theoremstyle{Remark}
%\newtheorem{remark}
\renewcommand{\today}{}
\newcommand{\abs}[1]{\left| #1\right|}
\newcommand{\Lap}[1]{\mathcal{L}\left\{#1\right\}}


\addtolength{\topmargin}{-.875in}
\addtolength{\textheight}{1.75in}

\newcommand{\solution}[1]{
\color{red}\begin{quote}Solution:\quad 
\color{black} #1
\end{quote}\color{black}
}
\newcommand{\ba}{\backslash}
\DeclareMathOperator{\Gcd}{gcd}
\renewcommand{\gcd}[2]{\Gcd\left(#1, #2\right)}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\floor}[1]{\left\lfloor #1\right\rfloor}
\newcommand{\C}{\mathbb{C}}
\DeclareMathOperator{\Diam}{diam}
\newcommand{\Mod}[1]{\ (\bmod\ #1)}
\newcommand{\diam}[1]{\Diam\left(#1\right)}


\begin{document}
\title{Math 127 Homework}
\author{Mary Radcliffe}
\date{Due 19 April 2018}
\maketitle

Complete the following problems. Fully justify each response.


\begin{enumerate}

\item Let $(X, 0, 1, +, \cdot)$ be a field, where $X$ is a finite set. Prove that there is no ordering $\leq$ on $X$ under which $X$ is an ordered field.

\item Let $(X, 0, 1, +, \cdot, \leq)$ be an ordered field. Prove each of the following basic ordered field properties, from axioms.
\begin{enumerate}
\item For all $x\in X$, $x^2>0$.
\item For all $w, x, y, z\in X$, if $w\leq x$ and $y\leq z$, then $w+y\leq x+z$.
\item For all $x, y, z\in X$, if $x\geq 0$ and $y\leq z$, then $xy\leq xz$.
\item For all $x, y, z\in X$, if $x\leq 0$ and $y\leq z$, then $xy\geq xz$.
\end{enumerate}

\item Let $\vec{x}, \vec{y}, \vec{z}\in \R^n$. Prove that $\|\vec{x}-\vec{z}\|\leq \|\vec{x}-\vec{y}\|+\|\vec{y}-\vec{z}\|$.

\item Let $x_n = \frac{n+2}{n+1}$. Prove that $x_n$ converges to $1$.

\item Let $(x_n)$ and $(y_n)$ be sequences of real numbers, with $(x_n)\to a$ and $(y_n)\to b$. Let $z_n=x_ny_n$ for all $n\in \N$. Prove that $(z_n)\to ab$.

\item Prove that if $(x_n)$ is a monotonically decreasing sequence, having a lower bound, then $(x_n)$ converges.


\end{enumerate}

\end{document}