Math Proof
Author:
Matthew Pelto
Last Updated:
před 10 lety
License:
Creative Commons CC BY 4.0
Abstract:
Proof
\begin
Discover why 18 million people worldwide trust Overleaf with their work.
\begin
Discover why 18 million people worldwide trust Overleaf with their work.
\documentclass[12pt]{amsart}
\pagestyle{empty}
\usepackage{amsmath,amssymb,amsfonts,amsthm}
\usepackage{enumerate}% http://ctan.org/pkg/enumerate
\usepackage{eucal}
\usepackage{graphicx,psfrag} %only include if using pictures
\usepackage{ifthen} %only include if using conditional package
%\usepackage{xymatrix}
%keep this...%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vfuzz2pt % Don't report over-full v-boxes if over-edge is small
\hfuzz2pt % Don't report over-full h-boxes if over-edge is small
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Side Margins
\evensidemargin 0.1 in \oddsidemargin 0.1 in
%Paragraph Size
\parindent 24pt
%size of page
\textheight 9.6 in \textwidth 6.2 in
\baselineskip 9.6 in \topmargin 0.005 in
% ***********************************************************************
\newcommand{\Normalstretch}{1.0} %change to define space between lines
\renewcommand{\baselinestretch}{\Normalstretch}
\newenvironment{singlespace}
{\renewcommand{\baselinestretch}{1} \small \normalsize}
{\renewcommand{\baselinestretch}{\Normalstretch} \small \normalsize}
% ***********************************************************************
\newcommand{\baseenvskip}{\baselineskip 2mm}
% Standard Notation for Theorems and Lemmas
\newtheorem{mydef}{Definition}
\newtheorem{thm}{Theorem}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{claim}[thm]{Claim}
\newtheorem{proposition}[thm]{Proposition}
\newtheorem{conjecture}[thm]{Conjecture}
\newtheorem{corollary}[thm]{Corollary}
\theoremstyle{remark}
\newtheorem{rmk}{Remark}[section]
\newenvironment{remark}{\begin{rmk}\rm\baseenvskip}{\end{rmk}}
\newtheorem{definition}[rmk]{Definition}
%How numbering is defined in articles (standard)
\numberwithin{equation}{section} \numberwithin{thm}{section}
\numberwithin{rmk}{section} \numberwithin{figure}{section}
% ************************ space ************************************
\newcommand{\hhs}[1]{\hspace{#1mm}}
\newcommand{\hs}{\hspace{5mm}}
\newcommand{\vp}{\vspace{1mm}}
\newcommand{\vs}{\vspace{5mm}}
\newcommand{\jl}{$\frac{}{}$} %User defined for empty symbol to jump line
% ********************** newcommand *********************************
\newcommand{\mbf}[1]{\mbox{\boldmath $#1$}}
\newcommand{\hb}[1]{\hspace{-#1 mm}}
\newcommand{\ds}{\displaystyle}
\newcommand{\QED}{\hfill $\Box$}
% *********************** frequently used math symbols from AMS *******
\newcommand{\norm}[1]{\left\Vert#1\right\Vert}
\newcommand{\abs}[1]{\left\vert#1\right\vert}
\newcommand{\set}[1]{\left\{#1\right\}}
% ************** Some frequently used symbols - User defined ***********
% ************** Also Called MACROS ************************************
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Hv}{\mathcal{H}_v}
\newcommand{\h}{\mathcal{H}}
\newcommand{\Ov}{\mathcal{O}_v}
\newcommand{\F}{\mathcal{F}}
\newcommand{\Rn}{R^n}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Ce}{\widehat{\mathbb{C}}}
\newcommand{\si}{\sigma}
\newcommand{\Cn}{\mathbb{C}^n}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\p}{\partial}
\newcommand{\ep}{\varepsilon}
\newcommand{\D}{\delta}
\newcommand{\eps}{\varepsilon}
\newcommand{\finv}{f^{-1}}
\newcommand{\im}{\imath}
\newcommand{\ga}{\gamma}
\newcommand{\ze}{\zeta}
\newcommand{\fee}{\varphi}
\newcommand{\noi}{\noindent}
%
% *******************************************************************
% At First run of Template, only modify from here below........ *****
% *******************************************************************
\begin{document}
\title{}
% ******************** ABSTRACT *************************************
%\begin{abstract}
%Insert abstract
%\end{abstract}
% Must be present so the above information is displayed.
% *******************************************************************
% **** Begin Typing your work from here below ***********************
% *******************************************************************
Let $K$ be a compact set in a metric space $(X,d)$. Suppose $\mathcal{F}=\{U_\alpha\}_{\alpha \in A}$ is an open cover of $K$, then there exists a positive number $\lambda$ so that for every $p \in K$ the open ball $B(p,\lambda)$ is contained in one of the open sets of $\mathcal{F}$.
\begin{proof}
Since $K \subset \underset{\alpha \in A}\cup U_\alpha$, for each point $p$ in $K$ there is a positive number $2\varepsilon(p)$ so that the ball $B(p,2\varepsilon(p))$ is contained in one of the open sets of $\mathcal{F}$. Clearly $\{B(p,2\varepsilon(p)\}_{p \in K}$ forms an open cover of K, and so by compactness this admits a finite refinement
\end{proof}
\end{document}