Lets say $a_1, a_n$ are normed vectors.  Why is the set $C = \{\Sigma_{i=1}^n \lambda_ia_i: \lambda_i \ge0\}$ closed? The $\lambda$'s can be any non-negative numbers. So C is the set of all non-negative linear combinations of the n vectors.  I tried both the method with open balls, and the method with a converging sequence(show that the point the series is convergin to is in C), but still I do not see why. (PS: If it is untrue in the generalized version, you can assume that the vectors are in $\mathbb{R}^n$.)
Looking for the proof of the lemma asserting that the conical surface (envelope) is a closed space. Thank you.
How to control a motor with only relays and push button? Hello I´ve been working in this project for some days, I need to make a circuit with only relays that can make a dc motor start, stop and a reverse. The thing is that when you push the start or the reverse button the motor need to keep running, I can´t keep it pushed.  This is as far as I can get. SW1 turns the engine on, SW5 turns it off and SW7 reverse the motor but only if it is already running. What i need is that the reverse button also turns the engine on. Please help.  notes: -i can only use three push buttons and as many relays as needed. Yes is a school assignment.  THANKS! For which positive integer values of $n$ does $a|mn-b?$ I am taking a number theory class, and I was not able to solve this homework problem.   Let $a,$ $b,$ and $m$ be constant positive integers. For which positive integer values of $n$ does $a|mn-b?$   I tried this out for a couple of values, and the values of $n$ satisfying this expression seem to occur in a cycle.  (For example, let $m=2019,$ $a=20,$ $b=19.$ The values of $n$ that work are $1,21,41,61,....$) However, I am not able to show that this always happens. Is there an easy way to calculate the total cost of tolls for a trip within the United States? Is there a convenient way to calculate the cost of tolls between two arbitrary points within the United States? I'm planning to take a trip along the Eastern seaboard and will be passing through multiple states with different toll-collecting authorities. The mapping and trip planning resources I've found so far will only indicate whether a particular section of road has tolls, not the price of those tolls. I'm aware that the various toll-collecting authorities post their rates online, but these charts are often difficult to use if you're not already familiar with the area. Isn't there an easier way? In a USB cable, is it OK to swap the D+ and D- wires? I heard that D+ and D- are differential signals, does it matter if I swap them when connecting a USB device to the computer? Please add TeX rendering on the Android app MathJax isn't currently working on the Android app, which makes it totally useless for the 'technical' sites like  or .  For example, in the Android app:    Same post in a (mobile) browser:     Please add this functionality! Are $_SESSION variables stored on the client or the server? How to send email from MySQL 5.1 "Communicate over" vs. "communicate through" Also schemas can communicate over database links to schemas in other remote databases.    Could one replace the preposition over with through? Will it change the meaning, and if so, when should we use over instead of through? I'm not quite sure how the total attack bonus interacts with BAB.   Can you make more attacks than shown by your BAB (e.g. +11/+6/+1 BAB allows you to make 3 attacks) by having a high enough DEX, STR or other attack bonus modifiers? Assuming I have +11/+6/+1 BAB, and a spell gives me +6 attack bonus, would that make me able to attack as if I had +17/+12/+7/+2 BAB, giving me an extra attack?)   Can you make multiple attacks even if you have one of your stats drained, making your attack bonus negative, or less than 5?      It says in the SRD that the formula for attack bonus in melee is this:       Base attack bonus + Strength modifier + size modifier   I've also been told that you can make multiple attacks in one turn, but each attack gains a -5 penalty to attack bonus. How to get the current date without the time? How to convert numbers between hexadecimal and decimal How do you convert between hexadecimal numbers and decimal numbers in C#? An example of a lattice that's not totally ordered Will upgrading Windows 8 to 8.1 affect Ubuntu? It's a rather simple question (I hope). I currently have Windows 8 and I am dual booting it with Ubuntu 13.04.   If I were to install the latest update for Windows 8, that being 8.1, will it affect Ubuntu or will there be any conflicts between the two? Add/Delete table rows dynamically using JavaScript I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it.  The Baire Category theorem asserts that if X is a complete metric space or a locally compact Hausdorff space (my case), then the complement of a countable union of nowhere dense sets is always nonempty.  How can I use this on the proof? How can I remove the BOM from XmlTextWriter using C#? How to show that when the linear transformation $\Phi =\psi \circ \phi : E \rightarrow G$ is injective, then $\phi$ is also injective? How to show that when the linear transformation $\Phi =\psi \circ \phi : E \rightarrow G$  (where $\phi : E \rightarrow F$ and $\psi : F \rightarrow G$) is (1) injective, then $\phi$ is injective and when (2) $\Phi$ is surjective, then $\psi$ is also surjective ?  I've tried to prove it by the definition of injection and surjection, but I'm not convinced myself by it.  For example, in the first case :  If $\Phi$ is injective, $\forall g \in G$, we have $0$ or $1$ $e \in E$ such as $\Phi(e)=g$.  $\Rightarrow \Phi(e) = \psi(\phi(e)) = \psi(f) =g$.   So $\forall f \in F$, we also have $0$ or $1$ $e \in E$ such as $\phi(e)=f \in F$.   $\Rightarrow \phi$ is injective.  Is this right or did I prove nothing at all ? I think it is easier to convince myself/ to understand with surjectivity but still.. Can't SSH in as root I'm using the default installation of openssh-server for my Ubuntu 14.04 server. I can connect through the console as root like normal, with my username and password. When I try to SSH with the same username/password, however, I repeatedly get Access Denied errors. /var/log/auth.log reports Failed password for root from &lt;ip address&gt; port &lt;port&gt; ssh2, but I'm entering the correct password.  Why can't I connect to the server via SSH, even though the username and password ARE correct? Two logos in beamer after certain frame REST API composite query: How do I use the result of a query subrequest in the referenceId of another subrequest?