Real analysis - inductive proof- help please
Proving $\prod \limits_{k=0}^{n}(1-a_k) \geq1- \sum\limits_{k=0}^{n}a_k$
I understand the idea of a focal length for a single lens, i e the distance from the lens to the point at which parallel rays of light converge. However, in the case of a photographic lens with multiple lens elements, where exactly is the focal length of the lens as a whole measured from? Proving the set of points at which a function diverges to $\infty$ is countable Let $f\colon\mathbb{R}\to\mathbb{R}$. Prove that the set    $$\{x \mid \mbox{if $y$ converges to $x$, then $f(y)$ converges to $\infty$}\}$$ is countable.   My book told me to consider  $g(x)=\arctan(f(x))$, then it said "it is easy to see the set is countable." But I still can't understand what it mean. My daughter is a US Citizen. We (parents)are also planning to permanently settle in US. We will go their as Permanent Resident,as per US Law. We have a house here in India worth Rs.2 Crores.  We want to sell it and take money with us. We have ordinary savings Bank account here in India.  My US Citizen Daughter is a Income Tax payer.  Now your advises are required as under:   What are the routes to take money from India to US and our tax liabilities Can I gift this whole amount to my US Citizen Daughter or what is the maximum limit of Gift amount.  What will be the tax liability on me and on my Daughter in case of Gift   Whether I have to show it in my Income Tax Return or in my Daughter's Tax Return. What US Income Tax Laws says. What will be the procedure to send money as Gift to my Daughter. Will the money will go directly from my Bank Act.to my Daughter's Bank Account. Will there will be wire transfer from bank to bank Can I send money through other money sender Certified Companies also.    Please advise. If there is something more on the subject to make me understand, please advise. Thanks, The link here:  shows that $\max(a,b)=\frac{a+b+|a-b|}{2}$.      Is it true that $\max(|a|,|b|)=\frac{1}{2}(|a|+|b|)$? change folder icons in ubuntu-mono-dark theme I use the grep command to search for a string in many files. Is there something similar to "search and replace"? Often floats may land quite a bit later in the document than the point they are created, sometimes after a section break.  Is there a way to force a new section to start on a new page, after any unplaced floats? Expected time to roll all 1 through 6 on a die Increasing the Size of Beamer Footer I work in Beamer extensively (specifically the Madrid theme). Sometimes, my "title" is quite long. Beamer puts this title at the bottom, but when it's too long it cuts it off. I'd really like to know how to increase the size of the bottom middle rectangular footer.    Here's the code (sorry for all of the packages and commands):  \documentclass[t]{beamer} %Add "mathserif" to script math lettering. "t" places all text to start at the top of each slide \usetheme{Madrid}  %Packages  \usepackage{amsmath} %Math mode \usepackage{amssymb} %Math mode \usepackage{commath} \usepackage{graphicx} %Including pictures \usepackage{color} %Define colours \usepackage{setspace} %Enables double-spacing/variable spacing \usepackage{pgfplots} %Draw graphs \usepackage{upgreek} %Insert theta symbol as unitalicized  \usepackage[version=3]{mhchem} %Chemistry equation mode \ce{} \usepackage[export]{adjustbox} %Place black borders around images \usepackage{scrextend} %Used to change font size \usepackage{caption} %Used to utilize * in caption function \usepackage{subfig} %Used to place figures side-by-side with sub-captions \usepackage{siunitx} %SI Units (+ scientific notation) \usepackage{cancel} \usepackage{fancybox} %\usepackage{enumitem} %Making lists %\usepackage[english]{babel} %Lists with letters  %Erases Beamer Navigation Symbols on the bottom left \setbeamertemplate{navigation symbols}{}  %\setbeamertemplate{footline}{} \captionsetup[subfigure]{labelformat=empty}  %Defined Colours \definecolor{orangered}{RGB}{255,69,0} \definecolor{darkgreen}{RGB}{0,100,0}  \definecolor{firebrick}{RGB}{178, 34, 34} \definecolor{navy}{RGB}{0,0,128} \definecolor{lightsteelblue}{RGB}{176,196,222} \definecolor{steelblue}{RGB}{70,130,180} \definecolor{darkslateblue}{RGB}{72, 61, 139} \definecolor{forestgreen}{RGB}{34, 139, 34} \definecolor{Gold}{RGB}{218,165,32}  %Defined Colour Theme and List Sizing \setbeamercolor{structure}{fg = Gold} \setbeamertemplate{navigation symbols}{}  \setbeamerfont*{itemize/enumerate body}{size=\normalsize} \setbeamerfont*{itemize/enumerate subbody}{parent=itemize/enumerate body} \setbeamerfont*{itemize/enumerate subsubbody}{parent=itemize/enumerate body}  %Opening Title Page \title{Chapter 7 - Absolute Value and Reciprocal Functions} \subtitle{Section 7.3 - Absolute Value Equations} \date{}  \begin{document}     %Slide 1     \maketitle \end{document} I have a problem when a lot of figures are in question. Some figures tend to "fly around", that is, be a paragraph below, although I placed them before that paragraph. I use code:  \begin{figure}[ht] \begin{center} \advance\leftskip-3cm \advance\rightskip-3cm \includegraphics[keepaspectratio=true,scale=0.6]{slike/visina8} \caption{} \label{visina8} \end{center}\end{figure}   to place my figures. How can I tell latex I REALLY want the figure in that specific place, no matter how much whitespace will be left? Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$  Apart from induction, I tried with  to check the validity, but I can't think of an easy (manual) alternative.  Please suggest an intuitive/easy method. What happens if a Rainbow Servant takes levels in Mystic Theurge? When a Rainbow Servant hits level 10 they gain access to all nondomain Cleric spells. All spells that do not appear on any other spell list they have are cast as Divine spells. This means that, if they have the skills and have learned 2nd level spells, they qualify for the Mystic Theurge Prestige Class.  But what happens when you actually start to take levels in it? The text says:     When a new mystic theurge level is gained, the character gains new spells per day as if he had also gained a level in any one arcane spellcasting class he belonged to before he added the prestige class and any one divine spellcasting class he belonged to previously.   What it does not say however, here or elsewhere, that these classes have to be different classes. Do you get one level in Wizard/Warmage/whaever for being an Arcane caster, then ANOTHER level for being a Divine caster as well? The text for the Rainbow Servant says the following:     Such spells are cast as divine spells if they don’t appear on the sorcerer/wizard or bard spell lists.   So it comes down to these questions:   Does being able to cast Divine spells qualify you as a Divine Caster, or is there more to it than that? If it is, what happens when you take levels in Mystic Theurge like this? Do you go up one level or two? What happens if this means you go over 20th caster level? Do you keep gaining spell slots like a level 20+ character, or does something else happen? Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$ I'm not sure how to go about this. Does it involve using MVT?  I got as far as saying $e = \frac{e^x - e^y}{x-y}$. A while back I read about the super Kamiokande detector detected a large neutrino flux and then several hours later a supernova was seen. Anyone know of this with sources? I don't recall the source at the moment. $\frac{1}{99989999} = 1.00010002000300050008001300210034005500890144... \times 10^{-8}$    (), which includes the Fibonacci sequence $(1\ 1\ 2\ 3\ 5\ 8\ 13\ 21\ 55\ 89\ 144\ldots )$.  This is fascinating to me but I can't figure out how this works, and a Google search doesn't seem to reveal anything. Shrink Tempdb log file when reaching a specific size how I can shrink tempdb log file when reaching a specific size?? I'm scheduling job to shrink tempdb but still I face a problem of disk space due to tempdb is full! Is it possible to activate an iPhone 4 without a SIM? Show that $(L^{p},\|\|_{p})$ is a Banach space.  My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So,  $(L^{1},\|\|_{1})$ is vector space. To show that is a Banach space, note that  Prop. 1: Let $(f_{n})_{n\in\mathbb{N}}\subset\mathcal{L}^{1}$ (where $\mathcal{L}^{1}=\mathcal{L}^{1}(X,\tau,\mu)$ is the space of all integrals function), such that $\sum_{i}{\|f_{n}\|}_{1}&lt;\infty$, then the sequences $\left(\sum_{n=1}^{N}{f_{n}}\right)_{n\in\mathbb{N}}$ converges almost everywhere to the integral function, that we called $\sum_{n\in\mathbb{N}}{f_{n}}$. Furthermore $$\sum_{n\in\mathbb{N}}{\int{f_{n}d\mu}}=\int{\sum_{n\in\mathbb{N}}{f_{n}d\mu}}\quad\lim_{N\to\infty}{\|\sum_{n=1}^{N}{f_{n}}-\sum_{n\in\mathbb{N}}{f_{n}}\|_{1}}=0$$ Prop. 2: Let $(E,\|\|)$ a normed vector space. Then $E$  is a Banach space if and only if for all $(e_{k})_{k\in\mathbb{N}}\subset E$ such that $\sum_{k\in\mathbb{N}}{\|e_{k}\|}&lt;\infty$, $\left(\sum_{k=1}^{n}{e_{k}}\right)_{n\in\mathbb{N}}$ converges in $E$  We take the proposition 2. Let $(f_{n})_{n\in\mathbb{N}}\subset L^{1}$ such that $\sum_{n}{\|\hat{f}_{n}\|_{1}}&lt;\infty$. Let $f_{n}\in\hat{f}_{n}$, then $\|f_{n}\|_{1}=\|\hat{f}_{n}\|_{1}$ and then $\sum_{n}{\|f_{n}\|_{1}}&lt;\infty$, by prop. 1, there exist $F\in\mathcal{L}^{1}$ such that $\|\sum_{n=1}^{N}{f_{n}-F\|_{1}}\to 0$, and then $\|\sum_{n=1}^{N}{\hat{f}_{n}-F\|_{1}}\to 0$. Finally note that $\hat{F}\in L^{1}$ is a Banach space. This was my answer for  $L^{1}$ space, but how I prove the general statement for $L^{p}$ (I want to do a similar response). Edit: I'm stuck in the problem, any idea or hint is appreciated.Thanks!! Get the coordinates from ContourPlot and RegionPlot How do I create RAW TCP/IP packets in C++? I'm a beginning C++ programmer / network admin, but I figure I can learn how to do this if someone points me in the right direction. Most of the tutorials are demonstrated using old code that no longer works for some reason.   Since I'm on Linux, all I need is an explanation on how to write raw Berkeley sockets. Can someone give me a quick run down?