Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms    $$p_{x}(f) = |f(x)|, \quad (0 \le x \le 1).$$   Show that there exists a function sequence $\{f_{n}\}$ in $X$ converging to $0$, such that $\{a_{n}\,f_{n}\}$ does not converge to $0$ for all sequences of scalars $\{a_{n}\}$ converging to $\infty$.   Hi everybody. This problem supports the point that the metrizability is necessary for $X$ to make this theorem true:      If a sequence of functions $\{f_{n}\}$ in a metrizable topological vector space $X$ converges to $0$, then there is a scalar sequence $\{a_{n}\}$ converging to $\infty$ such that $\{a_{n}\,f_{n}\}$ converges to $0$.   But I got no clue how to find that sequence like above. Also, topologize a vector space by family of seminorms makes me confused so much. So I hope someone can help me. Thanks.
This is a problem from Rudin's Functional Analysis:  Show that there is a sequence of functions $\{ f_n \}$ in $X$ (the vector space of all complex functions) such that $f_n \to 0$ pointwise, but if any sequence $\{ \gamma_n \}$ tends to $\infty$, $\gamma_n f_n \not\to 0$.  The hint says to use the fact that the collections of all complex sequences converging to $0$ has the same cardinality as $[0,1]$. Honestly, this confused me more than it helped me.  My main issue is through the method of showing existence: I am not sure if I should attempt to construct an explicit sequence $f_n$, or if I need to appeal to other nonconstructive means such as Zorn's Lemma, etc. I have been trying to construct clever choices of sequences for a good bit now, but it always seems like I will be able to find a sequence that diverges slowly enough to allow that $\gamma_n f_n \to 0$.   Thanks!
How to convert decimal to hexadecimal in JavaScript Earlier, I voted to close  as off-topic.  The question has since been closed.  There is an , but it didn't show up in the list when I selected "off-topic".  The options were:   Meta Stack Overflow Server Fault Super User Webmasters Programmers   Questions:   Should Wordpress appear in this list? Are there any other sites that should appear in the list?   Should all non-beta sites appear in that list by default? SEO scope for multiple domains hosting on the same IP address I use dd in its simplest form to clone a hard drive:  dd if=INPUT of=OUTPUT   However, I read in the manpage that dd knows a blocksize parameter. Is there an optimal value for the blocksize parameter that will speed up the cloning procedure? Pointwise convergent and total variation I'm preparing for a test for real analysis and I came across this problem in Royden's book:  Let $\{f_n\}$ be a sequence of real valued functions on $[a,b]$ that converges pointwisely on $[a,b]$ to the real valued function $f$. Show that $TV(f) \leq \liminf  ~TV(f_n)?$  This looks quite similar in form to Fatou's Lemma to me, but can't find any way to establish TV with integration, can anybody please help?  (TV is short for total variation) Deploy Community Builder audience page assignments with ant migration tool? In Community Builder, if you create audiences and assign them to page variations, you cannot deploy these using change sets, according to .  In fact, Audience doesn't even appear in the list of change set components.  On the other hand, Audience metadata can be deployed using the ant migration tool, so I was hopeful that the audience assignments would also be deployable.  Unfortunately, I haven't had success in deploying any audience assignments, either for entire pages or for components on a page.  Has anyone found a workaround for deploying audience assignments for Community Builder pages?  I should say that I'm using the Build Your Own template.  If it can't be deployed, is there at least a way to get a list of audience assignments so I don't have to open every single page in the Community Builder to check?  Here's what I'm using in my package.xml manifest.  All the community builder page variations and all the audiences are created correctly, but the assignments are not.  &lt;types&gt;     &lt;members&gt;*&lt;/members&gt;     &lt;name&gt;Audience&lt;/name&gt; &lt;/types&gt; &lt;types&gt;     &lt;members&gt;*&lt;/members&gt;     &lt;name&gt;CustomSite&lt;/name&gt; &lt;/types&gt; &lt;types&gt;     &lt;members&gt;*&lt;/members&gt;     &lt;name&gt;SiteDotCom&lt;/name&gt; &lt;/types&gt; I have a .bash_profile in my home directory, but it isn't getting run on login. If I do the following, then things seem to be as I expect:  ssh myhost bash source ~/.bash_profile   But normally that all happens on login. Thoughts? Let $A=\begin{pmatrix}0&amp;5\\ 7&amp;4\end{pmatrix}$.      1) Write $A$ as a product of $4$ elementary matrices.      2) Write $A^{-1}$ as a product of $4$ elementary matrices.   My work. I have managed to find $A^{-1}$, which came out to be this:  \begin{pmatrix}-\frac{4}{35}&amp;\frac{1}{7}\\ \frac{1}{5}&amp;0\end{pmatrix}  However, I am struggling to figure out how I would split each of these matrices into $4$ elementary matrices. Any help? Area of a weird ellipse shape. A propeller has the shape shown below. The boundary of the internal hole is given by $r = a + b\cos(4α)$ where $a &gt; b &gt;0$. The external boundary of the propeller is given by $r = c + d\cos(3α)$ where $c - d &gt; a + b$ and $d &gt; 0$.  Calculate the total area of the propeller as shown by the shaded region and calculate the average temperature of the propeller, given by $T (r, α) = C/r$, where $C$ is a constant.     Have absolutely no idea how to solve this. Why are my quotation marks causing an “unexpected token” error with this command? Is there any way to make a method with more than one instance? Laptop randomly waking up = Hibernate / Low Battery? Here is what I'd like to achieve in QGIS.   Is there a processing tool to remove the overlapping part of polygons?    It is also very important that they do not get merged during the process and that they keep their original attributes since I need to work with them afterwards. GPG can't check signature If you have the ability to cast a spell without a spell slot, can you cast that spell using a spell slot? A 10th level Drow warlock (for instance) has the following Drow Magic racial trait (PHB p. 24), in addition to its warlock spellcasting abilities:  You know the dancing lights cantrip. When you reach 3rd level, you can cast the faerie fire spell once per day. When you reach 5th level, you can also cast the darkness spell once per day. Charisma is your spellcasting ability for these spells.  Can the Drow warlock cast faerie fire (which is not on the warlock spell list) using a warlock spell slot? Can it cast darkness (which is on the warlock spell list) using a warlock spell slot?  The  says this about the Magic Initiate feat:  If you’re a spellcaster, can you pick your own class when you gain the Magic Initiate feat? Yes, the feat doesn’t say you can’t. For example, if you’re a wizard and gain the Magic Initiate feat, you can choose wizard and thereby learn two more wizard cantrips and another 1st-level wizard spell. If you have spell slots, can you use them to cast the 1st-level spell you learn with the Magic Initiate feat? Yes, but only if the class you pick for the feat is one of your classes. For example, if you pick sorcerer and you are a sorcerer, the Spellcasting feature for that class tells you that you can use your spell slots to cast the sorcerer spells you know, so you can use your spell slots to cast the 1st-level sorcerer spell you learn from Magic Initiate. Similarly, if you are a wizard and pick that class for the feat, you learn a 1st-level wizard spell, which you could add to your spellbook and subsequently prepare.  But this specifically requires you to pick the feat for one of your classes to be able to cast the spell learnt using your class spell slots, and does not seem directly applicable. A Warlock can use warlock Pact Magic spell slots to cast spells learnt from levels of another spellcasting class, per the multiclass spellcasting rules (PHB p. 164):  If you have both the Spellcasting class feature and the Pact Magic class feature from the warlock class, you can use the spell slots you gain from the Pact Magic feature to cast spells you know or have prepared from classes with the Spellcasting class feature, and you can use the spell slots you gain from the Spellcasting class feature to cast warlock spells you know.  Similarly, if the warlock has learnt the invocation Mask of Many Faces (PHB p. 112)...  You can cast disguise self at will, without expending a spell slot.  ...Can the warlock cast disguise self using a spell slot, e.g. to decrease the chance it can be dispelled? I have created 50 spreadsheets for 50 students. Each spreadsheet contains 33 sheets, numbered 1 to 33, along with a couple of other sheets.  Within each of these 33 sheets, there are many protected ranges.  I accidentally protected a range that ought not to have been protected. So, with 50 students each having 33 sheets, I have over 1800 sheets to unprotected.  I want to loop through each sheet, 1 to 33, and remove protection from cell F15. Right now, I (the owner) am the only editor.  Since I'm not an actual programmer, I've clearly not done this properly.  Can someone offer a suggestion of how I can change my Google Script to make this work?  var ss = SpreadsheetApp.getActiveSpreadsheet(); var sheets = ss.getSheets();  for (var i=2; i  &lt;sheets.length; i++) {   var sheet = ss.getSheets()[i];   sheet.activate();   SpreadsheetApp.flush();   var range = sheet.getRange('F15');        var protection = range.protect().setDescription('Protected range');        var me = Session.getEffectiveUser();        protection.addEditor(me);        protection.removeEditors(protection.getEditors());        if (protection.canDomainEdit()) {          protection.remove();          } //end of if statement  } Can I prevent others from editing my question? edited my question,  where I specifically said that our software sending back stacktrace to one of our servers was a user-known public feature.  He then modified the title of my post, along with the entire content, making it sound like I snuck in the feature and that our application was phoning home without our users knowing.    I specifically wrote in the original post that the feature was not sneaky. He completely changed the meaning of my post. I find this to be rather unacceptable.  A few questions:   Is it possible to prevent editing of my posts? How can I retaliate? Can I file a complaint?   Can people with  do anything they want on Stack Overflow? Send values from one form to another form Converting Raw HTTP Request into HTTPWebRequest Object I am using Adobe Illustrator CS6 to create small images (like icons) for my app so I need pixel precision.  How can I avoid X, Y, W and H properties to have decimals? Can I configure 0 decimals for this properties on my document?  If I use the textboxes with this properties it does right, it increases pixel by pixel, but when draggin with the mouse it uses up to 3 decimals.