Revision as of 22:12, 31 May 2024 by Bot (Created page with "<div class="d-none"><math> \newcommand{\indexmark}[1]{#1\markboth{#1}{#1}} \newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\NOTE}[1]{$^{\textcolor{red}\clubsuit}$\marginpar{\setstretch{0.5}$^{\scriptscriptstyle\textcolor{red}\clubsuit}$\textcolor{blue}{\bf\tiny #1}}} \newcommand\xoverline[2][0.75]{% \sbox{\myboxA}{$\m@th#2$}% \setbox\myboxB\null% Phantom box \ht\myboxB=\ht\myboxA% \dp\myboxB=\dp\myboxA% \wd\myboxB=#1\wd\myboxA% Scale phantom...")
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BBy Bot
May 31'24

Exercise

[math] \newcommand{\indexmark}[1]{#1\markboth{#1}{#1}} \newcommand{\red}[1]{\textcolor{red}{#1}} \newcommand{\NOTE}[1]{$^{\textcolor{red}\clubsuit}$\marginpar{\setstretch{0.5}$^{\scriptscriptstyle\textcolor{red}\clubsuit}$\textcolor{blue}{\bf\tiny #1}}} \newcommand\xoverline[2][0.75]{% \sbox{\myboxA}{$\m@th#2$}% \setbox\myboxB\null% Phantom box \ht\myboxB=\ht\myboxA% \dp\myboxB=\dp\myboxA% \wd\myboxB=#1\wd\myboxA% Scale phantom \sbox\myboxB{$\m@th\overline{\copy\myboxB}$}% Overlined phantom \setlength\mylenA{\the\wd\myboxA}% calc width diff \addtolength\mylenA{-\the\wd\myboxB}% \ifdim\wd\myboxB\lt\wd\myboxA% \rlap{\hskip 0.35\mylenA\usebox\myboxB}{\usebox\myboxA}% \else \hskip -0.5\mylenA\rlap{\usebox\myboxA}{\hskip 0.5\mylenA\usebox\myboxB}% \fi} \newcommand{\smallfrac}[2]{\scalebox{1.35}{\ensuremath{\frac{#1}{#2}}}} \newcommand{\medfrac}[2]{\scalebox{1.2}{\ensuremath{\frac{#1}{#2}}}} \newcommand{\textfrac}[2]{{\textstyle\ensuremath{\frac{#1}{#2}}}} \newcommand{\nsum}[1][1.4]{% only for \displaystyle \mathop{% \raisebox {-#1\depthofsumsign+1\depthofsumsign} \newcommand{\tr}{\operatorname{tr}} \newcommand{\e}{\operatorname{e}} \newcommand{\B}{\operatorname{B}} \newcommand{\Bbar}{\xoverline[0.75]{\operatorname{B}}} \newcommand{\pr}{\operatorname{pr}} \newcommand{\dd}{\operatorname{d}\hspace{-1pt}} \newcommand{\E}{\operatorname{E}} \newcommand{\V}{\operatorname{V}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Bigsum}[2]{\ensuremath{\mathop{\textstyle\sum}_{#1}^{#2}}} \newcommand{\ran}{\operatorname{ran}} \newcommand{\card}{\#} \newcommand{\Conv}{\mathop{\scalebox{1.1}{\raisebox{-0.08ex}{$\ast$}}}}% \usepackage{pgfplots} \newcommand{\filledsquare}{\begin{picture}(0,0)(0,0)\put(-4,1.4){$\scriptscriptstyle\text{\ding{110}}$}\end{picture}\hspace{2pt}} \newcommand{\mathds}{\mathbb}[/math]

\label{Pa0} Show that in Lemma indeed [math]a_0\in(1,\sqrt{5}-1)[/math] holds. \smallskip { \small Hint: Establish first that [math]\exp(x)\leqslant 1+x+x^2/2[/math] holds for [math]x\leqslant0[/math]. }