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Proof of "continuity from above" and "continuity from below" from the axioms of probability One of the consequences of the axioms of probability ($\sigma$ field and probability axiom) is the "infinite subset" and "infinite union" property, I can't figure out how it follows from them.  if $A_1 \subseteq A_2 \subseteq \cdots$ then   $$P\left(\bigcup_{i=1}^{\infty}A_i \right) = \lim_{k \to \infty} P(A_k)  $$  If $A_1 \supseteq A_2 \supseteq \cdots$ then   $$P\left(\bigcap_{i=1}^{\infty}A_i \right) = \lim_{k \to \infty} P(A_k)  $$ Drush CC all returning "No Drupal site found, only 'drush' cache was cleared" I've tried other solutions, but none of them are specific to my setup. I just started using the new Linux subsystem featured in the recent Windows 10 update. I'm using WAMP x64 v3.0.6, and have successfully cloned my website and got it up and running. At the root of the Linux subsystem, I installed Drush globally with Composer, using the directions given on the .  Typing drush status returns the following when I run it in my default directory:  Drupal version         :  7.50  Site URI               :  http://default  Database driver        :  mysql  Database hostname      :  localhost  Database port          :  Database username      :  root  Database name          :  aerosphere  PHP executable         :  /usr/bin/php  PHP configuration      :  /etc/php5/cli/php.ini  PHP OS                 :  Linux  Drush script           :  /home/bob/.composer/vendor/drush/drush/drush.php  Drush version          :  7.1.0  Drush temp directory   :  /tmp  Drush configuration    :  Drush alias files      :  Drupal root            :  /mnt/c/wamp64/www/drupal-7.50  Site path              :  sites/default   But, as mentioned in the title, drush cc all returns No Drupal site found, only 'drush' cache was cleared.[warning] I assume it has something to do with adding a path to the .bashrc, but I don't know what to add. Any tips would be greatly appreciated!  My question is different from "", they're on a Mac and using MAMP, and I'm using WAMP on Windows. I've tried the solutions offered on that post, but the issue has not been resolved.  I'm obviously having issues connecting to MySQL as I get this when I run mysql command:     The program 'mysql' can be found in the following packages: * mysql-client-core-5.5 * mariadb-client-core-5.5 * mysql-client-core-5.6 * percona-xtradb-cluster-client-5.5      Try: sudo apt-get install .   So I've added these lines to my .bashrc file:  PATH="$HOME/.composer/vendor/bin:$PATH" export PATH="/mnt/c/wamp64/mysql/mysql5.7.14/bin:$PATH"   The top line was added to get drush working. I've checked multiple times, and I have the path correctly set, but it's still not working. I'm searching a way to append a group in Python.  It should behave exactly like the Append function (ShiftF1) works, all objects should be local and editable. The idea is to append a character (with rig etc.) via Python and not Link it (to be able to modify it without making everything local one by one).  Thanks How can I download free apps without registering an Apple ID? If a topology contains all infinite subsets, then it is the discrete topology I am solving question $8$ of exercises in section 1.1. of chapter 1 in "Topology without tears". The question reads as follows.     Let $X$ be an infinite set and $\tau$ be a topology on $X$. If every infinite subset of $X$ is in $\tau$, prove that $\tau$ is the discrete topology   My idea was to show that every singleton set is in the topology and thereby the topology is discrete.  Choose an infinite set $A$ from $X$ such that $A^c$ is also infinite. $A$ and $A^c$ belong to $\tau$ since they are infinite sets. Now consider any $x \in X$. Note that $A \cup \{x\}$ and $A^c \cup \{x\}$ belong to $\tau$ since they are infinite sets. Hence, their intersection which is nothing but $\{x\} \in \tau$ for every $x \in X$.  The problem I have is I don't know how to prove the first sentence in the previous paragraph. These are my line of thoughts to prove them.   If $X$ is a countably infinite set, then I can list the elements are let the odd numbered elements fall into $A$. This guarantees $A$ and $A^c$ are both infinite. If $X$ is uncountable, then I can choose a countably infinite subset and call it $A$.   Are the above arguments rigorous? I am not especially happy with my second argument of choosing a countably infinite subset from an uncountable set since I do not give an explicit procedure of constructing the set $A$.  Also, is there any other simpler way of answering the original question? Password Reset for Users on a Multisite Subsite I have a multisite network. I want my users to be able to reset their passwords while staying within their subsites. So for example clicking the "lost your password?" link keeps them within their subdomain (subdomain.domain.com/wp-login.php?action=lostpassword) as opposed to taking them to the main site (domain.com/wp-login.php?action=lostpassword).  After they submit the username or email to reset the password, I would like for that email to contain links back to their subdomain and not the main site as is the default functionality.   Prior to the 4.0 update I was able to use this patch  and ultimately this plugin  to get the exact functionality described above. All was great.  However after the update it is no longer working. Asking for assistance in the various places referenced has still not helped solve the problem. Is anyone able to help? PHP XML how to output nice format The spell was cast without words, simply a flick of the Death Eater's wand, and a 'streak of purple flame' that lanced across her chest. It was later mentioned that the spell was weakened because of the lack of a verbal component, but still 'did enough damage as it was'. She was in the infirmary healing from the damage for some time. What spell was this? What is the difference between a static and a non-static initialization code block I have a program which I am trying to run, however when I run it; it just complains that it can't find a particular file.  However I have no idea which folder it is trying to find this particular file in. I have a copy of the required file, I just need to know which folder to copy it too.  Is there any way to show in real time which files are being accessed or which files are trying and failing to be accessed?  I am using Ext4 filesystem if that helps.  Thanks If I'm standing at the equator, jump, and land 1 second later, the  Earth does NOT move 1000mph (or .28 miles per second) relative to me,  since my velocity while jumping is also 1000mph.   However, the Earth is moving in a circle (albeit a very large one),  while I, while jumping, am moving in a straight line.   How much do I move relative to my starting point because of this? I  realize it will be a miniscule amount, and not noticeable in practise,  but I'd be interested in the theoretical answer. SqlServer: can one table's PK be the FK of two table's? I have three tables project, users and proj_user. The PK of proj_user table is the FK of project and users table's pk e.g.(project_id,user_id)  My question is, is it possible if the project_id and user_id's data is different  like project_id = '12345' and user_id = 'abc987'  Please reply  kind regards Vivek Why don't virtual particles violate conservation of mass/energy? "OpenID 2.0 for Google Accounts is going away"; will this affect Stack Exchange? How to flag a typo question I came across  today. In reading the answer and the comments it turns out that the issue was just a simple typo that the OP made. So that being the case I decided that this question probably needs to be closed since this will not be helpful to any future vistors. However, in looking at the flag reasons (I don't have the close ability yet), I don't see a flag that would be appropriate.   I know some of these options might still be influx because of the . However, until a different off-topic flag description is created, how should this be flagged? Or should this not be flagged at all and just allowed to live on or wait for a user with close votes to use a custom off-topic close option? Inequality of $\vert \text{tanh}(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$ I have an assignment, that I cannot seem to find a starting point for. I am not looking for the specific answer, but only where to begin. Any pointers are appreciated.  I have to prove the inequality $\vert \tanh(x)-x+\frac{x^3}{3}\vert\leq\frac{1}{8}$ for $x\in[-1/4,1/4]$ where $x-\frac{x^3}{3}$ is the 3rd order Taylor expansion.  I've tried with the triangle inequalities, Lagrange remainders, and a lot of other stuff, but I am no where near a correct argumentation. Does anyone have any suggestions? Uniform distribution on the surface of unit sphere Java generics super keyword I am thinking of buying a digital SLR camera. What things should I be looking for? Here is a question as phrased in Advanced Linear Algebra (Steven Roman):     Show that a linear operator $\tau$ on a finite-dimensional complex inner product space $V$ is normal if and only if whenever $S$ is an invariant subspace under $\tau$, so is $S^\perp$.   I'm only interested in the $\Rightarrow$ direction. Wikipedia (and ProofWiki)  that uses some kind of computation with the trace. Linear Algebra Done Right chooses an orthonormal basis and then uses the fact that $\Vert \tau v \Vert = \Vert \tau^* v \Vert$.  However, the above  proofs seem a bit too computational to me. Can the following argument work?  (Use $\odot$ for orthogonal direct sum.) Suppose that $\tau$ is normal and write $$V=E_{\lambda_1} \odot \cdots \odot E_{\lambda_k}$$ where the $E_{\lambda_i}$ are the eigenspaces of $\tau$. If $S$ is a $\tau$-invariant subspace then $\tau|_S$ is diagonalizable, so $$S=\widetilde{E}_{\lambda_1} \odot\cdots\odot \widetilde{E}_{\lambda_k}$$ where the $\widetilde{E}_{\lambda_i} \le E_{\lambda_i}$ are the eigenspaces of $\tau|_S$ (which could be trivial). Then choose orthogonal complements $D_i$ such that $E_{\lambda_i} = \widetilde{E}_{\lambda_i} \odot D_i$, and then $$S^\perp = D_1 \odot\cdots\odot D_k.$$ Since each $D_i$ is a subspace of an eigenspace, they are all $\tau$-invariant.