If $(x_α)_{α \in A}$ is a collection of numbers $x_α ∈ [0, +\infty]$ such that $\sum_{α∈A}{x_α} &lt; \infty$, show that $x_α = 0$ for all but at most countably many $α \in A$, even if $A$ itself is uncountable.
Let $A$ be an infinite set. For each $a \in A$, let $x_a$ be a non-negative number. Let the value of the series $\sum_{a \in A} x_a$ be the value of the supremum of its finite partial sums, i.e.:      $$\sum_{a \in A} x_a = \sup_{n \geq 0} \sup_{(a_1, \ldots , a_n) \subset A} \left(x_{a_1} + x_{a_1} + \ldots + x_{a_n}\right)$$   (Supremum is well defined even when its value is infinite)       If $\sum_{a \in A} {x_a}$ is finite, prove that the set $A' = \left\{{a \in A | a &gt; 0}\right\}$   is countable.   My guess is that we have to write $A$ as a union of finitely many sets.
How do you take off body armor in GTA Online? It's kind of hard to shop for clothes with a big bullet proof vest on.  Is there a way to take off the armor after missions, or at least hide it? A type-I superconductor can expel almost all magnetic flux (below some critical value $H_c$) from its interior when superconducting. Light as we know is an electromagnetic wave. So what would happen if we shine a light deep into a type-I superconductor? Suppose the magnetic field of the light is significantly lower than $H_c$.  And what about type-II superconductors? sf:INVALID_OPERATION: INVALID_OPERATION: testLevel of NoTestRun cannot be used in production organizations While deploying code in vsCode from sandbox to production I get the below error      sf:INVALID_OPERATION: INVALID_OPERATION: testLevel of NoTestRun cannot   be used in production organizations   As per the error, I need to change the value of testLevel but I am not able to find where and how can I change it to RunLocalTests  Could someone please help me to change the value of testLevel in vsCode Can magnets rotate infinitely? There are many videos on youtube in which people arranged magnets in circle and rotated one placing in middle of that circle on a shaft, and the magnet (magnet motor) starts madly and continues its movement. Do they really rotate infinitely? If so, do they from where they get that extra energy to move so fast and infinitely?    EDIT: This question is about real world scenario  UPDATE: I am unable to find that particular video about which i talked about -- I added them to my favourites, but they are all now deleted :( -- there are hundreds of that that kind of videos. Some example videos links patching SQL Server with Availability Group and Secondary Replica Accepting an answer should automatically upvote it as well Suppose k is an integer such that $6k + 1$, $12k + 1$, $18k + 1$ are primes. Show that $n = (6k + 1)(12k + 1)(18k + 1)$ is a Carmichael number. Suppose k is an integer such that $6k + 1$, $12k + 1$, $18k + 1$ are primes. Show that $n = (6k + 1)(12k + 1)(18k + 1)$ is a Carmichael number.  This is a question on a past exam that I find difficulty to answer. Can anyone help? What exactly is a kilogram-meter? This question isn't just about kilogram-meters, it's about multiplied units in general.  I have a good mental conception of divided units, e.g. meters per second or grams per cubic meter.  Meters per second just (usually) means how many meters are traversed by an object each second.  Simple.  But I don't really have a mental conception of what it means when units are multiplied, e.g. a kilogram-meter.  Can anyone explain in simple physical terms what a kilogram-meter is? Finding square root without using sqrt function? I've just finished running my first session with only one player, and it felt pretty different from running a session for a group.  Could someone with more experience in this talk about what one should do differently in that situation?  I'm not referring to mechanics like encounter-building, but just how to handle the change to the social dynamics. Showing that $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$ I need prove the following: ($\zeta(s)$ is the Riemann zeta function)  $\displaystyle\lim_{s \to{1+}}{(s-1)\zeta(s)}=1$  I really don't know, I have tried hard, but have obtained nothing for now. How do you get \mathbb{1} to work (characteristic function of a set)? Is there any shorter pronunciation of W than double-U? When spelling, every letter in the Alphabet is pronounced by a single syllable, with the only exception of W being pronounced "double-U". (Fun fact, in German it's approximately pronounced like the "ve" in very, but in some strike of cosmic justice Y is pronounced "Yps-e-lon") Why is that? Is there any at least semi-officially accepted one-syllable alternative pronunciation of W?  I checked some related post, e.g. , but they are more concerned with the "Why?" instead of the "Why not different?" Outlier Detection on skewed Distributions Under a classical definition of an outlier as a data point outide the 1.5* IQR from the upper or lower quartile, there is an assumption of a non-skewed distribution. For skewed distributions (Exponential, Poisson, Geometric, etc) is the best way to detect an outlier by analyzing a transform of the original function?  For example, distributions loosely governed by an exponential distribution, could be transformed with a log function - at which point is it acceptable to look for outliers based on the same IQR definition? What does functools.wraps do? The problem   I have tens of thousands of small polygons for which I want to create buffer/doughnuts for, but I don’t get the units right. I want meters and not degrees, and changing to (what I think is) the right CRS my layer doesn’t longer show. The polygons are small farming plots. I work in QGIS 3.4.15.   Let me take you through it  I start by adding a vector layer. I opt for ‘Directory’ as a source type (Encoding: UTF-8 by default) and under Source (Type: UK. NTF2 by default) I point to the folder in which I store the shapefile.   The default CRS is set to EPSG:4326 – WGS 84, for both layer and the project (checked in bottom right), and unit is thus in degrees. I add the layer world map (EPSG:4326 – WGS 84 by default) by taping “world” in the coordinate box in the middle bottom, I also add Open street map (EPSG:3857 – WGS 84 / Pseudo-Mercator by default) from XYZ Tiles. The polygons appear nicely on the map considering some are just adjacent to the nearby lake, many nicely adjacent to roads, etc.   When attempting to do buffers the distance is obviously set to degrees. I have tried to change the CRS, both for the layer(s) and the projects, but without luck. After researching (e.g. here: ) I believe that the correct CSR would be Arc 1960 / UTM zone 36S EPSG:21036 (it is my understanding that either 36N or 36S would work as the polygons appear both south and north of, but close enough to the equator). When setting the mentioned CRS for my polygon layer, the polygons disappear from the map, and remain gone even after setting the project CSR (by bottom right) to the same. It seems that whenever I try with different CRS in UTM, the layer doesn’t show. Does this mean that the polygons are somehow not compatible with units in meter? Is there an issue with importing the .prj file?  Other things I've tried:   I have made a good attempt searching for solutions. I have ensured that the polygon layer is on top of the layers list. I have ensured the file names are identical and tried all different suitable CRS I can think of. I have seen similar questions been solved by saving the .shp 'as', but I don't think that's my issue as it's already a .shp file. Can it be the case that I need to perform a datum transformation? I gave it a shot using a transformation code (1122) provided by above link, but then the polygons were projected on the Antarctica on the world map, so obviously something went quite wrong...   Note that when I have changed to what I believe is the right CRS I can execute buffers in units of meters, however, they do not appear on the window… Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+$ Find the sum of the following :   (i) $$\frac{1}{\cos0^\circ \cos1^\circ}+\frac{1}{\cos1^\circ\cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+\frac{1}{\cos88^\circ \cos89^\circ}$$  I tried :  $$\frac{1}{\cos1^\circ}\left[\frac{\cos(1^\circ-0^\circ)}{\cos0^\circ\cos1^\circ} + \frac{\cos(2^\circ-1^\circ)}{\cos1^\circ\cos2^\circ}+...\right]$$  = $$\frac{1}{\cos1^\circ}\left[\frac{\cos1^\circ\cos0^\circ}{\cos0^\circ\cos1^\circ} - \frac{\sin1^\circ \sin0^\circ}{\sin0^\circ\cos1^\circ} + \frac{\cos2^\circ \cos1^\circ}{\cos1^\circ\cos2^\circ} -\frac{\sin2^\circ \sin1^\circ}{\cos1^\circ\cos2^\circ}...\right]$$  For this, as well, I am not getting any pattern to solve further. Please suggest, thanks. "Contact us"... and then what? So I had a dispute with the moderators at a specific SE site. As a result, I got suspended there. As per SE guidelines, I pressed the "Contact us" button and wrote to the higher powers to appeal.  That was 5-6 weeks ago. There was zero feedback, neither positive, nor negative, not even "it's in progress, please wait".  So what is this "contact us" thing supposed to do? Is it a placebo to make me feel better while being routed to dev/null?  What is the normal time it takes to process a "contact us" message?  Notes:   As far as I remember, I did not contact the team before. I'm a longtime user of multiple SE sites and never had any problems except for that one site. I'm pretty sure my form was submitted. I didn't make screenshots or anything, because I did not think I would need them in a post like this. I didn't expect a super-fast reply either, so I wasn't wondering after a week or two. I work in D8, I create a custom module first_module. By modules create a url and display all nodes by *.html.twig file.  Write code in controller.php  public function contentthree() {    $connection = Database::getConnection();    $query = $connection-&gt;select('node', 'n');   $query-&gt;fields('n', ['nid', 'type']);    $groups = $query-&gt;execute()-&gt;fetchAll();    foreach ($groups as $group) {     $events[] = [$group-&gt;nid, $group-&gt;type];   }    return [     '#theme'  =&gt; 'first_module_display',     '#events' =&gt; $events,   ]; }   In *.module  function first_module_theme($existing, $type, $theme, $path) {    return [     'first_module_display' =&gt; [       'variables' =&gt; ['events' =&gt; NULL],       'template'  =&gt; 'first-module-list',     ],   ]; }  function _get_node_details() {    $output = 'Welcome D8';    $output .= 'Node Details';    return $output; }   and create templates first-module-list.html.twig and display data  {%for key, data in events %}  {%for key1, data1 in data %}     {{ data1 }}  {% endfor %}  {% endfor %}   work fine and showing data correct. But, my question is how to call function _get_node_details() and how to render node details in first-module-list.html.twig files? C Scanf suddenly stopped reading in values