If $X$ is limit point compact, i.e., every infinite subset of $X$ has a limit point in $X$,  and $f:$ $X\mapsto Y$ is continuous function, is $f(X)$ limit point compact?.
I'm looking for proof verification/help for the title question. Here is what I have now: Let $f:X \rightarrow Y$ be continuous with $X$ limit point compact. Let $V$ be an infinite subset of $f(X)$. Then we also have that $f^{-1}(V)$ is also an infinite subset of $X$. Since $X$ is limit point compact there is $x \in X$ such that $x$ is a limit point of $f^{-1}(V)$. Then $x \in \overline{f^{-1}(V)}$ by definition of the closure. Hence since $f$ is continuous, $$ f(x) \in f(\overline{f^{-1}(V)}) \subseteq \overline{f(f^{-1}(V))} \subseteq \overline{V} $$ Then since $f(x) \in \overline{V}$, it is either a limit point of $V$ or an element of $V$. However since is infinite, any neighborhood of $f(x)$ will contain infinitely many points of $V$ in either case. Thus $V$ has a limit point and $f(X)$ is limit point compact.
LINQ's Distinct() on a particular property Solving $\sin x\cos3x+\cos x\sin3x=\sqrt{3}/2$ in the interval $[0,2\pi]$ Solve the following trig equation: $$\sin(x)\cos(3x)+\cos(x)\sin(3x)=\frac{\sqrt{3}}{2}$$ in the interval $[0,2\pi]$.  Using trig identities, it is now at $$\sin(x+3x)=\sqrt{3}/2 \tag{1}$$ Next, it has become $$\sin(x+3x)=\sin 60^\circ \tag{2}$$ leading to $$\sin(x+3x)-\sin60^\circ=0 \tag{3}$$ From there, it is unclear what to do. I am interested in displaying some things (e.g.: title page) in full color on screen, while having the same object/page display in a black and white friendly form when printed.  According to , there is built-in functionality in the PDF specification for Screen-Only objects and Screen-And-Print objects.  There may also be a way to add a layer on top of the B&amp;W page that only displays the full color page on screen via the PDF 1.5 Optional Content Groups feature.  Similarly,  demonstrates the selective printing of text via ocg-p.  But it is not clear how I can use this package in the way that I need to.  My scenario is compounded by the fact that my color page text is white on a dark background.  MWE:  This code presents the way I am currently switching back and forth between color and b&amp;w.  I change the printincolor bool variable to make the change.  \documentclass[11pt]{article}  \usepackage[demo]{graphicx} \usepackage{etoolbox}  \providebool{printincolor} % Will set color title page and section headings \setbool{printincolor}{false}  % Setup color if needed \ifbool{printincolor}{% \usepackage[pagecolor=none]{pagecolor}% \definecolor{MyBlue}{HTML}{1b3667}% \usepackage{afterpage}}{}  \begin{document}  % Begin Title Page {  \thispagestyle{empty}   \ifbool{printincolor}{%   \pagecolor{MyBlue}%   \afterpage{\nopagecolor}}{}   \begin{center}   {\ifbool{printincolor}{\color{white}\sffamily}{\sffamily}%  Set text color white or leave [default] black   \ifbool{printincolor}{\includegraphics[width=1.5in]{logo-gold}}{\includegraphics{logo-blue}}% put logo gold or blue   \\ \vskip 1in   \begin{huge}         \textbf{Title}         \vskip 0.1ex\rule{0.5\textwidth}{0.8pt}\vskip 1ex         Subtitle         \\         \end{huge}   \vfill\vfill   \begin{large}    Authors:\\   John Doe and Jane Doe\\   \end{large}   }   \end{center} }  \end{document}   How can I layer (or otherwise display) two versions of the title page in a way that color displays on screen, but black and white is printed? 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[1]   [2] shortcut for commands I use following commands very often, and they're too long.  sudo /etc/init.d/kerio-kvc start sudo /etc/init.d/kerio-kvc stop   how can I create a shortcut command for them? something like sudo kerio start or sudo kerio stop I know that the usual ordering of $\mathbb R$ is not a well-ordering but is there an uncountable $S\subset \mathbb R$ such that S is well-ordered by $&lt;_\mathbb R$?   Intuitively I'd say there is no such set but intuitively I'd also say there is no well-ordered uncountable set at all, which is obviously wrong. I still struggle to grasp the idea of an uncountable, well-ordered set. how to make some border lines of a table thick and colored Convergence in measure - product I'm trying to prove the following statements in Folland's book.  Let $(X,\mathcal{M},\mu)$ be a measure space. 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