(Analysis 1 by Tao) Exercise 8.1.8 Use Corollary 8.1.13 to prove Corollary 8.1.14. Corollary 8.1.13. The set $\mathbb{N} \times \mathbb{N}$ is countable. Corollary 8.1.14. If $X$ and $Y$ are countable, then $X \times Y$ is countable.  For the proof of Corollary 8.1.13, the book shows that $\mathbb{N} \times \mathbb{N} = A \cup B$, where $A = \{(n,m) \in \mathbb{N} \times \mathbb{N} : 0 \le m \le n\}$ and $B = \{(m,n) \in \mathbb{N} \times \mathbb{N} : 0 \le n \le m\}$. I also know that $X$ and $Y$ are countable, there exists bijections from $\mathbb{N}$ to $X$ and $\mathbb{N} $ to $Y$. To finish the proof, I need to find the bijection from $\mathbb{N} \times \mathbb{N}$ to $X \times Y$, but I do not know how to get this. Any help would be appreciated.
The Cartesian product of a family $(A_i\mid i\in I)$ is defined as $$\prod\limits_{i\in I}A_i=\{f:I\to\bigcup A_i\mid f(i)\in A_i\}$$ Let $I_n=\{i \in \mathbb N \mid i \le n\}$ for $n\in \Bbb N$ and $A_i$ be countable for all $i\in I_n$. Then $\prod\limits_{i\in I_n}A_i$ is countable.   My attempt: Lemma 1: $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ and $\prod\limits_{i\in I_{n+1}}A_i$ are equinumerous for all $n \in \mathbb N$. (I presented a proof ) Lemma 2: If $A$ and $B$ are countable, then $A \times B$ is countable. (I presented a proof ) I will prove the theorem by induction on $n$. The theorem is trivially true for $n=0$. Assume that the theorem is true for $n=k$, then $\prod\limits_{i\in I_k}A_i$ is countable. Since $\prod\limits_{i\in I_k}A_i$ is countable (by inductive hypothesis) and $A_{k+1}$ is countable, then by Lemma 2 $(\prod\limits_{i\in I_n}A_i) \times A_{n+1}$ is countable. Furthermore, by Lemma 1 $(\prod\limits_{i\in I_k}A_i) \times A_{k+1}$ and $\prod\limits_{i\in I_{k+1}}A_i$ are equinumerous, then $\prod\limits_{i\in I_{k+1}}A_i$ is countable too. Thus the theorem is true for $n=k+1$. This completes the proof.   Does this proof look fine or contain gaps? Do you have suggestions? Many thanks for your dedicated help!
Using the Well Ordering Principle to Show that n^3 - n is divisible by 6. I read the following two sources and I think I understand the structure of proofs involving WOP, but I am stuck on the open-ended part where you prove P(n-1) for the  following problem:    Use the well-ordering principle to show that n^3 - n is divisible by 6.    Any help would be greatly appreciated!   Note: After factoring out (n-1)^3 - (n-1) it took the form of n^3 - 3n^2 + 2n and the only difference between that and n^3 - n is "3n^2 - 3n" which I proved was divisible by 6 via induction. Is there a better way to solve this problem?  Edit: , post is useful however I not supposed to use induction for the main part of the proof. The use of imaginary and complex values comes up in many physics and engineering derivations. I have a question about that: Is the use of complex numbers simply to make the process of derivation easier, or is it an essential ingredient, without which it would be impossible to derive some results?   I can identify two different settings where the answer may be different:   It doesn't look like it is mandatory for results in Newtonian mechanics, General Relativity and classical electrodynamics. Can we say the same thing about quantum mechanics either way for sure?   Could this be a difference in quantum mechanics over the classical picture? Why do high resolution cameras shoot stills at high resolutions, but only typically shoot video at 1080p which is about 2 megapixels?  For example I have a Sony-NEX 6 which can shoot stills at 16 megapixels but only 1080p by 1920p video.  Also how do digital cameras separate the frames in video if the lens shutter never closes?  And if you can shoot video at 30 fps or 60 fps then why can't cameras have a continuous drives at 30 to 60 fps? How to export a individual layer with a fixed master layer in Photoshop? Let's say I have some layers and a single different one, or a group of layers I want to export merged together, in a way that every single exported layer is merged with a specific layer or group of layers.  In the example below I have some layers which need to be exported into JPG or PNG, merged with the very top layer called "Blurred". In my case, I want to export the layers (Jan 8 + Blurred) and (Layer 10 + Blurred) and so on. Acting as exporting layers merged with a master layer called "Blurred".  My question in a simple way: how to apply a Master layer(s)/Group(s) in Photoshop to be exported with every layer I have? Where do I find out how a command/environment is defined? Say I want to redefine the itemize environment, or make a modified version of it. I want to know how it is currently defined. Where do I find this information? If it's some particular package command, I can just look at the .sty file for that package, but if it's something more basic, I don't know where to look...  Is there a reference manual of where all this stuff is explained? Or is there a file in the depths of my texmf tree I should poke around in? Prediction based on bayesian model I have created a bayesian model that estimates 6 parameters using rjags from R. Now i want to do some predictions based on new data in R. Can anyone help me with an example.  model {    B.m[1] &lt;- Kb # mean B[1]    B[1] ~ dnorm( B.m[1], tau_B)    for(t in 2 : 10) { # Process model       B.m[t]&lt;- max( B[t-1] + rB[t-1](1 -B[t-1]/K) - C[t-1] , 0.0)       B[t] ~ dnorm(B.m[t], tau_B) } # Observation mode       for (t in 1: 10) {          LogC_mean[t] &lt;- log( max(qsegB[t]Eff[t], 0.000001))          C[t] ~ dlnorm(LogC_mean[t], tau_c) } # Prior          b~dbeta(1, 1)          cv &lt;- 1          K.mean &lt;- 500000          K.v &lt;- pow(cvK.mean, 2)          K.tau &lt;- 1/K.v          K ~ dnorm(K.mean, K.tau)          r ~ dlnorm(0.0, 1.4)          qseg ~ dlnorm(-15.4, 1.44)          tau_c ~ dgamma(0.001,0.001)          tau_B ~ dgamma(0.01,0.01)        }     } Where can I send feature requests? How do I check if an integer is even or odd? USB flash drive not working or is appearing as an empty disk drive, Disk Management reports "No Media" with 0 bytes size Imgur is over capacity - some user avatars fail to load I see a lot of 'white' avatars in the question lists, e.g.  and . Some user avatars aren't loaded, e.g.  and  – they just timeout for me. Sometimes (but not always) I do get a nice "Imgur is over capacity!" image, which I'm (rather surprisingly) able to upload to imgur - but you might not see it. Getting raw SQL query string from PDO prepared statements Is there a way to get the raw SQL string executed when calling PDOStatement::execute() on a prepared statement? For debugging purposes this would be extremely useful. How to make a teleport command that does not work in the Nether? I have a vanilla server 1.8.7 that uses a book to tp players to spawn or to their home (an invisible armor stand with their uuid connected to it). Problem is, if they set home (place their armor stand) in the overworld and go to the nether and try to tp home (activated using a trigger), it sends them to the same co-ordinates in the nether, and most likely kills them.   I did some digging, and there seems to be a dimension parameter, but I can't quite figure out the syntax or how to get it to work.  I updated the command block to:  /tp @a[score_tpHome_min=1,Dimension=0] 68 68 68   I was hoping it would only tp the player when they were in the overworld (Dimension 0), but it seems to be ignoring that parameter in the nether and still teleporting to those co-ordinates in the nether.   I also tried:  /tp @a[score_tpHome_min=1,Dimension:0] 68 68 68   Which is the way other people were using the parameter in other ways, but that gave an error. What should children call a step-grandparent? My mom remarried to a great guy who has been absolutely wonderful to my kids. Ever since my mom and he were dating my kids have called him Mr. Joe. Lately my mom has been bugging me to get the kids to call him Grampa.  I agree that he deserves more respect than Mr.  The problem is that he isn't their real grampa so I don't really want them calling him that. Does anybody have any suggestions on what we should have the kids call him? They are 9 and 12 years old.  EDIT: I appreciate the comments regarding different variations of grampa and hearing how others have no issues with calling their step-granddad grampa. But I was hoping maybe someone has some suggestions for names that show the importance of the step-granddad without calling him some variation of the grampa name but still has a feeling of warmth and family in the name (I hope that makes sense). Why are there 2 packages for some applications (e.g. Chromium) in Ubuntu Software? What is the difference between French and British cuts of beef? What is the difference between French and British cuts of beef?   I am told they just butcher the animals dfferently. Certainly the cuts don't seem the same. For example is faux fillet really exactly the same as British sirloin and is entrecôte really the same as rib steak?  Here is a picture of British beef cuts. Sci-fi book, blue critters than only some people can see, kids fight the critters as they feed off energy and are a threat I can't really remember how long ago it was, but I think the book was marketed towards children/teens.  All I remember is the basic premise of the story.  Some story about these unusual little blue electric critters that only certain people could see, and these kids took it upon themselves to go about destroying them since they fed off energy or something like that.  Not sure whether they fed off people or off electricity... or both maybe?  I think the dominant method of destroying them was an electric baton that was taken from law enforcement.  I wanna call them buzz batons, but that's probably thanks to Artemis Fowl. Javascript close alert box I want to be able to close an alert box automatically using Javascript after a certain amount of time or on a specific event (i.e. onkeypress).  From my research, it doesn't look like that's possible with the built-in alert() function.  Is there a way to override it and have control over the dialog box that it opens?  Also, I don't want an override that shows a hidden div as the alert.  I need an actual dialog box. How to remove time from date? Database program with CSV import and ad-hoc SQL queries? Is there a database program for the Mac that makes it easy to import a CSV file into a table and then query and manipulate the data using ad-hoc SQL queries?  (A Windows example of a similar product would be Microsoft Access ... it also has forms and reports, which are not as important, but I would not mind a Mac solution that has those too.) Redirecting stdout to a file you don't have write permission on When you attempt to modify a file without having write permissions on it, you get an error:  &gt; touch /tmp/foo &amp;&amp; sudo chown root /tmp/foo &gt; echo test &gt; /tmp/foo zsh: permission denied: /tmp/foo   Sudoing doesn't help, because it runs the command as root, but the shell handles redirecting stdout and opens the file as you anyway:  &gt; sudo echo test &gt; /tmp/foo zsh: permission denied: /tmp/foo   Is there an easy way to redirect stdout to a file you don't have permission to write to, besides opening a shell as root and manipulating the file that way?  &gt; sudo su # echo test &gt; /tmp/foo