Definition: A set $A$ is countable if it is finite or if there is a bijection $c: \mathbb N \to A$; otherwise it is uncountable.     Let $I_n = \{i\in\mathbb N\mid i \leq n\}$. Then $\prod\limits_{n \in \mathbb N}I_n$ is uncountable.   This question arose when I tried to prove the above theorem.  I found that there exists an injection $\begin {array}{l|rcl} f : &amp; \mathbb N  &amp; \longrightarrow &amp; \prod\limits_{n \in \mathbb N}I_n \\     &amp; n &amp; \longmapsto &amp; f(n) \end{array}$  $f(n)$ is defined by $f(n)(i)=\begin{cases}i &amp; \text{ if } i&lt;n\\ n &amp; \text{ if } i \ge n \end {cases}$  I'm not sure if there is a surjection from $\mathbb N$ onto $\prod\limits_{n \in \mathbb N}I_n$?
Let $\{E_n\}_{n\in\mathbb{N}}$ be a sequence of countable sets and let $S=E_1\times\cdots\times E_n\times\cdots $. Show $S$ is uncountable. Prove that the same statement holds if each $E_n=\{0,1\}$.  By the definition of Cartesian product of sets,   $$\displaystyle S=\Pi_{n\in\mathbb{N}} \{f\colon\mathbb{N}\rightarrow\cup_{n\in\mathbb{N}}E_n\mid\forall n, f(n)\in E_n\}$$  If $E_n=\{ 0,1\}$, then  $$\displaystyle S_{01}=\Pi_{n\in\mathbb{N}}\{0,1\}=E^{\mathbb{N}}$$, where $E=\{0,1\}$.  By a theorem, $\cup_{n\in\mathbb{N}} E_n$ is countable since the sequence is countable.  I'm not sure how to go on from here to show $S$ is uncountable. Can we say anything about the function $f$ that maps a countable $\mathbb{N}$ to another countable union of sequence of countable sets?
I have a fresh installation of Ubuntu 14.10 on my laptop. It appears that letters are missing from files and folders, as well as menus and configuration windows. I'm not sure where to start in order to fix this. I just ordered the Nvidia GTX card. I have a dilemma, though. Should I keep using the driver which is available in "additional drivers" in Ubuntu, or should I install the driver from the Nvidia site?  So which driver is the best for me? Physics history book with some math I am looking for a book explaining physics from, say, Galileo and Newton till now; the book should be written using some math, similarly to my physics books when I was at the university (Halliday Resnik). I am searching a book that take care of explaining the true "evolution" of reasoning, discoveries, experiment to prove or disprove something.  I do not like a book that gives definitions without explaining what had happened. I also like to find, on that book, explanations on how actual measurements are taken (how can we measure the speed of light, the mass of celestial bodies, the attributes of electron, and how were the measurements taken in the past - how was possible to measure the speed of light in 1700-1800?).  I am not very interested in recent theories like strings, branes.... so it is OK if these are not discussed.  I like to find a lot of discussions on concept that are related, like for instance the implications on space dimensionalities, ether non-existence, strange facts, missing explanations. Why am I still getting a password prompt with ssh with public key authentication? I'm working from the URL I found here:    My ssh client is Ubuntu 64 bit 11.10 desktop and my server is Centos 6.2 64 bit. I have followed the directions. I still get a password prompt on ssh.  I'm not sure what to do next. How to avoid returning a Null if there is no "else" condition in an If construct I've just gotten through replacing capacitors on a trio of dead LCD screens (nothing's blown up yet, so far) - they either had one or two capacitors on their inverter circuit SLIGHTLY bloated, and not quite leaky. I ended up replacing all capacitors of the same brand/'colour', even the ones that looked fine, in case.  Now, checking a bad resistor is simple - i can use a standard multimeter to test it, and i tend to check my solders with the continuity testing option of the multimeter.  How would i test a capacitor ? Is there some standard, common way to test one? How to prove the following product? $$\frac{\sin(x)}{x}= \left(1+\frac{x}{\pi}\right) \left(1-\frac{x}{\pi}\right) \left(1+\frac{x}{2\pi}\right) \left(1-\frac{x}{2\pi}\right) \left(1+\frac{x}{3\pi}\right) \left(1-\frac{x}{3\pi}\right)\cdots$$ I don't understand the fact that let's say hypothetically I was pushing a block on a frictionless surface, and I kept applying the same force on the object how it would keep accelerating! Wouldn't there have to be a certain velocity where it would stop getting faster? If not then what is actually acting on the object to cause it to accelerate infinitely? What is NDSolve`FEM`*? Unicode input retrieved via PrimeFaces input components become corrupted A matrix inside a table Possible Duplicate:          I am trying to insert a matrix inside a table, and I am looking for a good way to do this. Usually when I write matrices in equations, I use array environment, so I naively defined an array inside tabular, and it sort of works. The only problem is that the matrix itself seems to take up a huge amount of space inside the cell. An example is   \begin{tabular}{|c|c|} \hline \textbf{EKF} &amp; \textbf{UKF}\\ \hline $Q = \left[ \begin{array}{cc} 0.1 &amp; 0 \\ 0 &amp; 0.1 \end{array}\right]$ &amp; $Q = left[ \begin{array}{cc} 0.01 &amp; 0 \\ 0 &amp; 0.01  \end{array}\right]$\\ \hline $R = 5 \times 10^{-3}$ &amp; $R = 5 \times 10^{-5}$ \\ \hline \end{tabular}   Is there any way to increase the spacing in the cell containing the matrix without that matrix taking up all the space? I want to simplify the following:  $\sum_{n=0}^x 1 - (1 - I[a \ne 0] * I[a \ge b]) * (1 - I[b \ne 0] * I[b \ge c]) * (1 - I[c \ne 0] * I[c \ge d]) * (1 - I[d = 0])$ Short story or novel from the 1970s or before about two warring alien tribes UITableView cells strangely disappearing Do I need a swap partition? I have 4GB of RAM and I want to install Ubuntu. Do I need a to have a swap partition? Because my computer has 4GB RAM, I do not think it is necessary to have a swap partition. If I do need to have a swap partition, should it be a 4GB or an 8GB partition? Prove that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ converges absolutely I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis. In section 2.7, he has the following exercise:  Exercise 2.7.5 (a) Show that if $\sum{a_n}$ converges absolutely, then $\sum{a_n^2}$ also converges absolutely. Does this proposition hold without absolute convergence?  I'm posting about this here because my answer to that last question about whether the proposition holds without absolute convergence is "Yes", but my suspicions are raised by him merely asking the question. Usually questions like this are asked to point out that certain conditions are necessary in the statement of propositions, theorems, etc. I just want to see if I'm missing something here.  Anyway, here's how I prove the absolute convergence of $\sum{a_n^2}$, and note that I never use the fact that $\sum{a_n}$ is absolutely convergent:  Proof: Let $s_n = \sum_{i=1}^n{a_n^2}$. I want to show that $(s_n)$ is a Cauchy sequence. So, let $\epsilon &gt; 0$ and $n &gt; m$, and consider, \begin{equation}   \begin{aligned}     |s_n - s_m| &amp;= |(a_1^2 + a_2^2 + \cdots a_n^2) - (a_1^2 + a_2^2 + \cdots a_m^2)|\\                 &amp;= |(a_{m+1})^2 + (a_{m+2})^2 + \cdots + a_n^2|\\                 &amp;= |a_{m+1}|^2 + |a_{m+2}|^2 + \cdots + |a_n|^2\\   \end{aligned} \end{equation} Now, since $\sum{a_m}$ converges, the sequence $(a_m)$ has limit 0. Therefore I can choose an $N$ such that $|a_m| &lt; \sqrt{\epsilon/n}$ for all $m &gt; N$. So for all $n &gt; m&gt; N$,  we have, \begin{equation}   \begin{aligned}     |s_n - s_m| &amp;= |a_{m+1}|^2 + |a_{m+2}|^2 + \cdots + |a_n|^2\\                 &amp;&lt; \left(\sqrt{\frac{\epsilon}{n}}\right)^2 + \left(\sqrt{\frac{\epsilon}{n}}\right)^2 + \cdots + \left(\sqrt{\frac{\epsilon}{n}}\right)^2\\                 &amp;&lt; \left(\sqrt{\frac{\epsilon}{n-m}}\right)^2 + \left(\sqrt{\frac{\epsilon}{n-m}}\right)^2 + \cdots + \left(\sqrt{\frac{\epsilon}{n-m}}\right)^2\\                 &amp;= \epsilon   \end{aligned} \end{equation} Therefore $(s_n)$ is Cauchy and $\sum{a_n^2}$ converges. And since $\sum{a_n^2} = \sum{|a_n^2|}$, the series $\sum{a_n^2}$ is absolutely convergent. &#x220E;  Am I doing something wrong here? Am I right in thinking that $\sum{a_n}$ need not be absolutely convergent for $\sum{a_n^2}$ to be absolutely convergent? How to plot and solve the numerical solution of a integro-differential equation Making the background of an image transparent in Gimp I have some jpg images and I need for each of them to have a transparent background.  I have GIMP as a graphic tool.   I need to put them in a web page and I need to see just the main element and have a transparent background. Prove Uncountable set minus a countable set is uncountable Let B be a countable set, and let A be an uncountable set. Prove that A-B is uncountable How do I create a Windows 7 USB installer with preinstalled software? I was wondering if it was possible to make a bootlable image of a Windows 7 Pro OEM Installation CD that I have and put it on an USB stick for faster/easier installation. I am asking because we have a couple of PC´s where I´d have to install the OS using the same key (MAK).  Also is it possible to kind of populate that image with software that would have to be installed anyways on all machines (like Chrome, Outlook, etc)?  It would be really helpful to just install a Windows 7 image with customized software already installed. I would just make an image of an OS and use that, but the hardware of the machines differ.