Let $J_n=\{1,\dots,n\}$. How do I show that the set of all functions $J_n\to \mathbb N$ is countable? Any function is given by specifying the images of $1,\dots,n$. There are $|\mathbb N|$ options for the image of each $i=1, \dots, n$. So intuitively, the set of such functions is the union of $n$ copies of $\mathbb N$, hence countable. But how to formalize it?
While I was reading Enderton's "A mathematical introduction to Logic", I came across the proof of the following sentence: "The set of all finite sequences of members of the countable set A is also countable".   Proof: The set S of all such finite sequences can be characterized by the equation  $$S=\bigcup_{n \in N} A^{n+1}$$ Since A is countable, we have a function f mapping A one-to-one into N. The basic idea is to map S one-to-one into N by assigning to $(a_0,a_1,...,a_m)$ the number $2^{f(a_0)+1}3^{f(a_1)+1}\cdot ... \cdot p_m^{f(a_m)+1}$, where $p_m$ is the $(m+1)$st prime. This suffers from the defect that this assignment might not be well-defined. For conceivably there could be $(a_0,a_1,...,a_m)=(b_0,b_1,...,b_n)$, with $a_i$ and $b_j$ in A but with $m\neq n$. But this is not serious; just assign to each member of S the smallest number obtainable in the above fashion. This gives us a well-defined map; it is easy to see that it is one-to-one.  Note: P is a finite sequence of members of A iff for some positive integer $n$, we have $P=(x_1,...,x_n)$, where each $x_i \in A$.  First of all, I cannot understand why the former assignment might not be well-defined and the latter assignment is well-defined. Secondly, I cannot understand what Enderton means by "just assign to each member of S the smallest number obtainable in the above fashion". By the way, is $(a,b,c,d) = ((a,b),(c,d))$ true? Also, in which cases can I omit/add parentheses in a tuple so as to have an equal tuple?
Formula for the $n$th term of $1, 2, 2, 3, 3, 3, 4, 4 ,4, 4, 5, ...$ Why are there two ticker symbols for Google — GOOGL and GOOG?  They seem to be mostly identical, but their market values are different: currently, 551.64 for  and 548.79 for . Android cancel Toast when exiting the app and when toast is being shown How do I reset an old iPod? My daughter gave her little brother her iPod Touch 8g. I have been trying to update the software and would like to get it rest to a factory clean state so I can linked it to the iTunes that already syncs to his iPad.   How can I do a clean up and wipe of everything from an iPod? The strange thing about the maximum in Planck's law Get mp3 duration in android More Generating Functions problems What system monitoring tools are available? I am looking for system monitoring tools which are GUI and CLI or web-based which include basic functions such as:   CPU Usage Ram Usage Swap Usage Disk Usage ( Space / I/O ) Heat Monitoring   I know there are many tools I can use, but I am looking for a single tool that has these basic functions. My phone was just stolen an hour or so ago. Are there any options for tracking? I realize this is a long shot, but my phone was lost or stolen a few hours ago.  At the time it was stolen, I didn't have any specific applications installed to track it or wipe the data. So... my bad. I'm hoping the fact that I have a pattern lock will at least prevent anyone from accessing my data.  My question is, do I have any options for tracking or data wipe after the fact? (I'm more interested in tracking.)  Note: I have read similar questions, but unless I'm mistaken, they all address the question in terms of preparation before losing the phone. I'm asking about fallback options after losing the phone. I have 14" 16:9 notebook and would like my Windows 10 taskbar have on the (left) side. Unfortunately, although I set it to small icons the bar is pretty wide. Do not you know how to make it thinner?  This is how I would like to have it: Mesh generation from PointCloud? The figure bleow is the pointcloud imported as .ply file into blender.   However, I would like to perform triangulation on it to produce the surface as seen in picture below(Accomplished using ballpivoting surface reconstruction and poisson disk sampling in MeshLab).   Is there a way to perform triangulation on belnder.  (PS: The purpose is to be able to select each faces/vertices individually to automate further.)  Edit: Question: Is it possible to select the vertex/faces programmatically post triangulation? if so, please explain how? Plotting an epicycloid Stirling's Approximation Proof I was searching for the reason why Stirling's Approximation holds true. I found the website  which apparently shows why this is the case. Does this part of the equation make sense in the proof? If so why?  $$\int_{1}^{n}\ln(x)dx\approx\sum_{k=1}^{n}\ln(k)$$ How do I install the Tor Browser Bundle in Ubuntu? In Ubuntu 13.04 I used to use the ppa:upubuntu-com/tor64  PPA to install the 64-bit Tor Browser Bundle. However, in Ubuntu 13.10 it doesn't work anymore. How do I install the Tor Browser Bundle in Ubuntu? I was just awarded +100 reputation on all of my Stack Exchange accounts.  What is this bonus for? It simply says "Association Bonus" on my reputation overview.  Also:   What privileges does the association bonus grant users? Can I lose the association bonus? Does it make my reputation 101 everywhere? Can I earn the association bonus on sites I haven't joined yet? Does it matter which site I earn the 200 reputation on? Will I get an association bonus every time my reputation on a site crosses 200 points? Does the association bonus allow me to answer protected questions? Does the association bonus count towards the reputation cap? Can I get the association bonus if I only have a profile on one site? What else is there to know about the association bonus? For example:   For an MVC architecture, should I prefer Ruby on Rails or PHP [closed] Gmail: What is a jQuery selector for incoming mail senders only? How to work with highly compressed and nested CSS in general? [closed]   Better to convert to   [closed] For an MVC architecture, should I prefer Ruby on Rails or PHP [closed] Gmail: What is a jQuery selector for incoming mail senders only? How to work with highly compressed and nested CSS in general?   I should first read the title and sometimes open the question to understand if the question is closed.  I think the [closed] mark  should be at the beginning of the title, not in the end, where I won't ever be if the title is long.  Also coloring will be a good improvement (via @Shadow Wizard). Can a spell scroll be upcast using a higher-level spell slot? For example,  says:  You create three glowing darts of magical force. Each dart hits a creature of your choice that you can see within range. A dart deals 1d4 + 1 force damage to its target. The darts all strike simultaneously, and you can direct them to hit one creature or several. At Higher Levels. When you cast this spell using a spell slot of 2nd level or higher, the spell creates one more dart for each slot level above 1st.  If a PC finds a scroll of Magic Missile, can it be cast as you would with a level 2 slot (and what are the conditions to do this)? Or will it always be level 1? Why isn't .profile sourced when opening a terminal? Just installed Ubuntu 14.04.1 LTS.  According to .profile,  # ~/.profile: executed by the command interpreter for login shells.  # This file is not read by bash, if ~/.bash_profile or ~/.bash_login exists.   There is no ~/.bash_profile or ~/.bash_login.  It does not seem to source .profile when opening a terminal.   Why doesn't it source .profile when opening a terminal? Why is .profile there if it is not sourced? How do I know/control what is sourced when a new terminal is opened? When a new terminal is opened, does it do a "login"? Where is a good place to add $HOME/bin to my path? .bashrc? Will that cause a problem because .profile has the same code? How to get the native version of Spotify running? Finding the limit of $(1-\cos x)/x^2$ $$\lim _{x \to 0}{1-\cos x\over x^2}={2\sin^2\left(\frac{x}{2}\right)\over x^2}={\frac{2}{x^2}\cdot {\sin^2\left(\frac{x}{2}\right)\over \left(\frac{x}{2}\right)^2}}\cdot\left(\frac{x}{2}\right)^2$$  now $$\lim_{x \to 0}{\sin^2\left(\frac{x}{2}\right)\over \left(\frac{x}{2}\right)^2}=\left({\sin\left(\frac{x}{2}\right)\over \left(\frac{x}{2}\right)}\right)^2=1^2$$  So we have $$\frac{2}{x^2}\cdot \left(\frac{x}{2}\right)^2=\frac{2}{x^2}\cdot \left(\frac{x^2}{4}\right)=\frac{1}{2}$$  Are the moves right?