I'm following a proof in Jech's book that every well ordered set is isomorphic to a unique ordinal and hitting a point where I'm not sure why a certain move is justified. He writes      Proof. The uniqueness follows from Lemma 2.7. Given a well-ordered set $W$,    we find an isomorphic ordinal as follows: Define $F(x) = \alpha$ if $\alpha$ is isomorphic    to the initial segment of $W$ given by $x$. If such an $\alpha$ exists, then it is unique.    By the Replacement Axioms, $ F(W) $ is a set. For each $x \in W$, such an $\alpha$ exists (otherwise consider the least $x$ for which such an $\alpha$ does not exist). If $\gamma$ is the least $\gamma \not\in F(W)$ then $F(W) =\gamma$ and we have an isomorphism of $W$ onto $\gamma$.   I've filled in all the details of this proof for myself up to the point where he assumes the class of ordinals not in the image of the function has a least element. Since this isn't a set, I'm not sure what can justify this. I could try to prove that every nonempty class of ordinals has a least element but that seems so substantial that it's hard to imagine it was a detail intentionally omitted. Am I missing something simple?
I am trying to solve this exercise from Jech's book on set theory:  Ex. 3.6: There is a well ordering of the class of all finite sequences of ordinals such that for each $\alpha$, the set of all finite sequences in $\omega_{\alpha}$ is an initial segment and its order-type is $\omega_{\alpha}$. Definition: If $W$ is a well ordered set and $u \in W$, then the set $\{ x \in W \mid  x &lt; u \}$ is an initial segment in W  Any help?
Is there a easy way to test ubuntu on new notebooks before buying it? write some AMPscript to skip a record if its empty I have a data extension with a CustomerId(PK) that has multiple Productid's as Productid1, ProductId2, ProductId3...ProductId10.   Some of the Productid's are empty for some Customers and I'm trying to write some AMPscript if the productIdx is empty to show the next one.  I'm trying to display 6 products in an email and at the moment when I preview the email, if from the first six Products one doesn't have a value(is empty) I get the error "The subscriber preview failed to generate. Review the details, correct all issues, and try again. Product1 not found"  I want to skip the product that is NULL and go to the next one available.  %%[ SET @customerId = [customerId]  SET @customer_rows = LookupRows("PRL","customerId",@customerId) IF Rowcount(@customer_rows) &gt; 0 THEN SET @customer_row = ROW(@customer_rows, 1)  SET @product_count = 0  SET @productId1 = FIELD(@customer_row, "productId1")  IF NOT EMPTY(@productId1) THEN SET @product_rows = LookupOrderedRows("PI_UK", 1,"Normal_SalePrice Desc", "ProductCode", @productId1)  IF RowCount(@product_rows) &gt; 0 THEN  SET @product_count = ADD(@product_count,1) ELSE  SET @product_count = ADD(@product_count,0) ENDIF ELSE SET @product_count = ADD(@product_count,0) ENDIF    SET @productId2 = FIELD(@customer_row, "productId2") IF NOT EMPTY(@productId2) THEN SET @product_rows = LookupOrderedRows("PI_UK", 1,"Normal_SalePrice Desc", "ProductCode", @productId2) IF RowCount(@product_rows) &gt; 0 THEN SET @product_count = ADD(@product_count,1) ELSE SET @product_count = ADD(@product_count,0) ENDIF ELSE SET @product_count = ADD(@product_count,0) ENDIF  SET @productId3 = FIELD(@customer_row, "productId3") IF NOT EMPTY(@productId3) THEN SET @product_rows = LookupOrderedRows("PI_UK", 1,"Normal_SalePrice Desc", "ProductCode", @productId3) IF RowCount(@product_rows) &gt; 0 THEN SET @product_count = ADD(@product_count,1) ELSE SET @product_count = ADD(@product_count,0) ENDIF ELSE SET @product_count = ADD(@product_count,0) ENDIF  and so on till ProducId10...   I have uploaded a photo with the PRL Data extension How can I sound proof bathroom walls? I'm trying to figure out the best, least intrusive way of sound proofing the walls of a bathroom. The bathroom shares a wall with my bedroom, and my 3 yea old's bedroom. The main problem is the sound of water running (plus every bump on the counter and everything) echos loud and clear through the wall into my sons room.  I found  and it seems like a good choice. Another option is adding another layer fo drywall with something like  between them. But all the walls and ceilings in the house are textured. So replacing just one or two walls in a room is just not possible.   This is the layout of the two rooms. I've added red box around the wall with the most noise.   Is there anything else I'm missing here that I could do to dampen the noise short of tearing the bathroom apart? Resistor in series needed with LED string? How to install Redis on Ubuntu 16.04? I am working with Laravel but i have installed all on my own (php, mysql, composer, nginx) and now i need to install Redis and configure it so i could use it for queue driver in Laravel.  How to install it, because it cannot be installed through apt-get install redis  i get this error:  ubuntu@ubuntu:~$ sudo apt-get install redis [sudo] password for ubuntu:  Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package redis I know references are syntactic sugar, so code is easier to read and write. But what is the difference between a pointer variable and a reference variable? Let $S=\{\{x\}\mid x \in \mathbb{R}\}$. Is $\sigma(S)$ included in the Borel $\sigma$-algebra on $\mathbb{R}$? Is $\sigma(S)$ equivalent to the Borel $\sigma$-algbrea on $\mathbb{R}$?    I think the answer to the first question is yes. Because the Borel sets include singletons, the Borel $\sigma$-algebra must contain the smallest sigma-algebra of the set of all singletons.   Here's my attempt at the second question: my answer would be no. Countable unions (or intersections) of countable sets are countable, so that any $X \in \sigma(S)$ is either countable or a compliment of a countable set. Therefore, the interval $(0,1) \not \in \sigma (S)$ but it is included in the Borel $\sigma$-algebra. My issue here is that I don't know how to formalize this statement, if it is even correct. Jordan normal form and invertible matrix of generalized eigenvectors proof Struggling to find a place to start with this proof- just began learning about Jordan normal.  Given a 2-by-2 matrix $A$ and a Jordan normal form matrix $J_{\lambda}$, there exists a matrix $S = [v1, v2]$ s.t. $$S^{-1}AS = J_{\lambda}$$ That is, $J_{\lambda}$ is similar to $A$.  WTS: $$(A - \lambda I)v_{1} = 0 $$ $$(A - \lambda I)v_{2} = v_{1}$$    I was going to try to start with the similarity condition and work into producing the two statements below.  Transform the similarity condition to: $$(A = SJ_{\lambda}S^{-1}) * v_{1}$$  Rewrite our first statement to prove as: $$Av_{1} = \lambda I v_{1}$$   Substitute: $$\lambda I v_{1} = SJ_{\lambda}S^{-1}v_{1}$$  But I've been continually getting stuck.  Tried a few ways other than this.  What intuition am I missing? $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$ The Wiki page on  says     The pair $(m, m+2)$ is twin prime, iff $4((m-1)! + 1) \equiv -m \pmod {m(m+2)}$.   This is obviously connected to Wilson's Theorem. Can anybody provide a proof for that, along the lines of thought given here, ? Fingerprint activation in Dell Vostro 14 in Ubuntu 18.04 I have Dell Vostro 14 with OS Ubuntu 18.04.  I tried   sudo apt install -y fprintd libpam-fprintd   but not able to use fingerprint reader of my laptop. Bathroom vanity light with no junction box Comparison of backup tools This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here.  While you are encouraged to help maintain its answers, please understand that "big list" questions are not generally allowed on Ask Ubuntu and will be closed per the .   Backup is incredibly important. Obviously there's no best backup tool, but a comparison of the options would be very interesting.   Graphical Interface? Command line? Incremental backups? Automatic backups? Install method: In standard repositories? PPA? Geometric solution to find a Set of totally ordered Subsets of $ \mathbb{N} $ is uncountable My question is, does there exist a set of subsets of the naturals, $S$, such that $S$ is uncountable, where for any two subsets $A,B \in S$ we have either $A \subset B $ or $B \subset A $?  Initially my thoughts were that we could construct a chain $A_{1} \subset A_{2} \subset \cdots $ and find a bijection by picking out one element which is 'new' to each set by axiom of choice and so we have bijected to a subset of naturals, however I'm not convinced this fully works.  Furthermore, I was thinking there may be a nice elegant solution by looking for such a set in either $ \mathbb{Q}^2 $ or $ \mathbb{Z}^2 $ for example, but any attempt seems to fail due to the 'ordering' part of the question. Forms Data Manipulation In Google Sheets I'm trying to develop a solution using Google Forms with Google Sheets that will collect Google Forms data and then automatically divide that one form submission's data into multiple rows of data in another sheet.   I've learned how to QUERY data (thanks to many of you on here) from one sheet to another, but I don't know how to automatically divide 1 row of data into multiple rows.  Here is a  in a visual manner (I will not delete it).   Basically, I have one Google form that collects the following.    Teacher Information (header data) Report Name (header data) Class Information (header data) Student#1 Data (record data: Multiple Form Fields) Student#2 Data (record data: Multiple Form Fields) Student#3 Data (record data: Multiple Form Fields) Student#4 Data (record data: Multiple Form Fields) Student#5 Data (record data: Multiple Form Fields)   I need an automated way to convert that single form response (1 row) into multiple rows (5 rows) each with the same 'header data' but with unique record date.   EXAMPLE Form Data  Header Data  -  Record Date (Student #1)  -  Record Date (Student #2)  -  Record Date (Student #3)  -  Record Date (Student #4)  -  Record Date (Student #5)   EXAMPLE Desired Manipulated Data      Header Data  -  Record Date (Student #1)     Header Data  -  Record Date (Student #2)     Header Data  -  Record Date (Student #3)     Header Data  -  Record Date (Student #4)     Header Data  -  Record Date (Student #5) Prove $p_n(x) \rightarrow \sqrt{x}$ uniformly as $n \rightarrow \infty$ Define $(p_n)_n$ recursively by $p_0(x) = 0$ and $p_{n+1}(x) = p_n(x) + \frac{1}{2}(x - p_n(x)^2)$.  Prove that $p_n(x) \rightarrow \sqrt{x}$ uniformly for $0 \leq x \leq 1$ as $n \rightarrow \infty$.  I'm not sure where to go with this.  I can show inductively that $\forall n,\  0 \leq p_n(x) \leq 1$ for $0 \leq x \leq 1$, and I think that $p_n(x)$ is increasing with $n$, but I have no idea whether this helps me.  I know that I need to show that $\forall \epsilon&gt;0: \exists N \in \mathbb{N}: \forall n \geq N: \forall x\in [0,1]: \big|p_n(x) - \sqrt{x}\big| &lt; \epsilon$ Edit:  I just had a thought.  I'm given a version of Dini's theorem that reads  Let $X$ be a compact metric space, $(f_n)_n$ a sequence of real-valued continuous functions on $X$, $f$ a continuous function on $X$ such that: (i) $f_n \rightarrow f$ pointwise on $X$, and (ii) $f_n(x) \geq f_{n+1}(x) $ for all $x \in X, n \in \mathbb{N}$. Then: $f_n \rightarrow f$ uniformly on $X$.  Obviously, (ii) is not true for my sequence, but what if I took $f_n(x) = -p_n(x)$ and $f(x) = - \sqrt{x}$?  Then I could prove (ii) to be true, I know that $[0,1]$ is compact, and $f_n(x)$ is real-valued.  The only part I would need to prove is (i), which is where I get confused.  Correct me if I'm wrong, but after that I could say that since $(-p_n)_n \rightarrow -\sqrt{x}$, then $(p_n)_n \rightarrow \sqrt{x}$.  Any advice? Will database replication increase the size of the replicated database? First thank you for your time. We have a SQL Server database that was assigned to be replicated. the original size of the database was 5 GB for the datafile, not the log file. Now after the replication the data file is 4 times bigger around 19 GB. Any pointers to see what could happened will be appreciated. I know sql as developer and some little management but nothing about replication. The performance of the database has been damaged very bad so I am trying to see what need to be done to improve it. I do apologize for my English as well. I have gridded depth points in CSV format that I'm trying to add other map data to, in additional columns. I've tried joining fields, hoping that I could link the Shapefile data to the csv, but when I try to display the csv by graduated symbols based on the new fields, it goes blank. I used Nathan's tips from , but imported the csv as csv since I want to add map info to that, rather than adding csv info to a map.  See this picture below for the sea bottom temperature &amp; salinity voronois by month, underneath the csv points (a sample subset), with the joins already made. I added columns on the csv for each of the fields i'm intending to append.    I also tried spatial joins as per  however it says the layers aren't the same CRS (I checked, they are), then gets to 15% and hangs.  I also also tried spatial join through  but it hit a python error. I can try this again and look to debug the problem if this is the right way to go.  In short: appending polygon data as fields on a points-grid csv: what's the best way in QGIS 2.0? robots meta not blocking indexing We have a staging version of our website to test changes on at trailheadpaddleshack.ca/staging1. This never appeared in search before. Recently the staging site has appeared on Google and is affecting our search results.  I'm trying to figure out how the pages got there and how to remove them. The pages have always had &lt;meta name="robots" content="noindex, nofollow"&gt; in the head.  I am kinda new at this but was under the impression this should prevent Google from showing my site in results. I am pretty sure the results appeared in google after I accidently copy pasted some codes from the staging site to the live site that contained links to pages on the staging site. If anyone can point me in the right direction to figure out what happened and prevent it from happening again would be much appreciated.  robots.txt looks like this:  User-agent: * Disallow: /calendar-2/action~posterboard/ Disallow: /calendar-2/action~agenda/ Disallow: /calendar-2/action~oneday/ Disallow: /calendar-2/action~month/ Disallow: /calendar-2/action~week/ Disallow: /calendar-2/action~stream/ #Begin Attracta SEO Tools Sitemap. Do not remove sitemap: http://cdn.attracta.com/sitemap/4035112.xml.gz #End Attracta SEO Tools Sitemap. Do not remove   I also tried adding an X-Robots-Tag header and submited the site to be re-crawled. Did that a few days ago and I still see no changes. Here are the HTTP headers according to "Fetch as Google":  HTTP/1.1 200 OK X-Robots-Tag: noindex,nofollow Vary: Accept-Encoding Transfer-Encoding: chunked Date: Sun, 24 May 2015 16:26:49 GMT Server: LiteSpeed Connection: close X-Pingback: http://trailheadpaddleshack.ca/staging1/xmlrpc.php Content-Type: text/html; charset=UTF-8 Link: &lt;http://trailheadpaddleshack.ca/staging1/?p=170&gt;; rel=shortlink   I am now faced with a bunch of results I need to remove from Google asap as they contain out of date information and are affecting our search results. Webmaster Tools has something for removal of a single URL but I am looking to remove the entire /staging1/ subfolder. Any tips? Error java.lang.OutOfMemoryError: GC overhead limit exceeded How do you handle "that that"? The double "that" problem