I am very bad with proofs that ask you to show that "$\exists$ something..." because in most of them you have to explicitly show the something. The following question is one such. Any help will be greatly appreciated!  Let $\{u_n\}$ a sequence in a Banach space $X$. Suppose that $\sum_{n=1}^{\infty} \|u_n\| &lt; \infty$. Prove that there exists some $x \in X$ such that  $$\lim_{n \to \infty} \sum_{k=1}^{n} u_k = x$$  Thank you very much in advance!
Does someone know if the following is true:  Let $\mathbb{X}$ be some arbitrary Banach space. $\{x_k \}_{k=1}^{\infty} \in \mathbb{X}$ is a sequence chosen from $\mathbb{X}$.  Now, if the series  $$\sum_{k=1}^\infty \|x_k\|_X$$ converges, would the "more generic" series $$\sum_{k=1}^\infty x_k$$  also converge?  If yes, could you please give the proof (or just mark the proof steps) ?  Thank you.
Show $$\frac{(2n)!}{n!\cdot 2^n}$$ is an integer for $n$ greater than or equal to $0$.   Could anyone please help me with this proving? Thanks! An $n \times n$ matrix $M$ of integers has the property that $M_{ab} = M_{xy}$ iff $a+b=x+y$. How many distinct elements are there in $M$? How to convert std::string to LPCWSTR in C++ (Unicode) The set of real numbers and power set of the natural numbers I have learnt that the cardinality of the power set of the natural numbers is equal to the cardinality of the real numbers. What is the function that gives the one-to-one correspondence between these two sets?  I have also learnt that there exists no set whose cardinality is strictly between the natural numbers and the real numbers. Is there a proof of this or at least some intuitiveness behind it? Are there any rules on the positioning adverbs should take in a sentence? How to snap and align 2 objects Hi! I'm a beginner learner and I'd like to know how to snap these three objects. I'd like to snap the two cylinders with the different faces of the cube. The short one on the left and the highest cylinder on the top of the cube. I'd like to have the center of the cylinder aligned with the center of the cube (the base of the cylinder inscribed into the face of the cube)  I'm sorry for bad explanation I hope someone can help me  Thanks!   I add a screenshoot in which you can see what I have to do   Thanks again if someone can help me with this! Expressions - is there an easy and "visual" way to combine them into another expression? If $(a,b)=1$ then prove $(a+b, ab)=1$. Let $a$ and $b$ be two integers such that $\left(a,b\right) = 1$. Prove that $\left(a+b, ab\right) = 1$.   $(a,b)=1$ means $a$ and $b$ have no prime factors in common  $ab$ is simply the product of factors of $a$ and factors of $b$.  Let's say $k\mid a+b$ where $k$ is some factor of $a$.  Then $ka=a+b$ and $ka-a=b$ and $a(k-l)=b$.  So $a(k-l)=b, \  a\mid a(k-1)$ [$a$ divides the left hand side] therefore $a\mid b$ [the right hand side].  But $(a,b)=1$ so $a$ cannot divide $b$.  We have a similar argument for $b$.  So $a+b$ is not divisible by any factors of $ab$.  Therefore, $(a+b, ab)=1$.  Would this be correct? Am I missing anything? In terminal emulation applications, pressing CTRL + Left / Right arrows jumps from one word to the previous or next one. Is it possible to have the same functionality in a Linux console, whether it is in text or in framebuffer modes?  In my configuration, the CTRL + arrow keys are transformed into escaped character sequences and not interpreted. How can I split an equation over two (or more) lines How to *disable* automatic reboots in Windows 10? Floating point comparison Is Gauss's law wrong, or is it possible that $\int_s{\vec E} \cdot d\vec{s}=0$ does not imply $\vec E = 0$? This is a question from David J Griffith's Introduction to Electrodynamics.      A specified charge density $\sigma(\theta)=k\cos(\theta) $ is glued over the surface of a spherical shell of radius $R$. Find the resulting potential inside and outside the sphere.   The question was solved using legendre polynomials and the final answer for potential inside the sphere was : $V(r,\theta) = \frac{kr}{3\epsilon_0}cos\theta$  This final answer is confusing because The Electric field inside the sphere is coming out to be dependent on $r$ and $\theta$ whereas electric field inside a shell, no matter what the charge distribution is outside, is $zero$ from gauss's law.      my doubts:         why is the electric field inside non zero?    Can gauss's law explain this, or does it fail here?      Since solving using ordinary surface integral gave me the same result and since the. divergence inside the shell is $zero$, I concluded that legendre polynomials and gauss's law in differential form are correct. So the problem should be with the integral form of gauss's law : $\int_s{\vec E}.d\vec{s} = \frac{q}{\epsilon_0}$  The answer I got for this doubt is that  "since the charges are glued to the surface, and not evenly distributed, electric field inside need not be zero. "     This is not convincing because proof of gauss's law does not expect the charges to be free to move. presence of an exrernal force that would hold the charges in place does not change the theorem. That is    Say only a single charge $q_i$ is present outside      then $\int_s{\vec E_i}.d\vec{s} = \frac{q_{inside}}{\epsilon_0}=0$      Now if there are more charges, following any distribution, net electric field $\vec E = \vec E_1 + \vec E_2+\vec E_3+...$       So the net flux,       $\int_s{\vec E}.d\vec{s} = \int_s{\vec E_1}.d\vec{s}+\int_s{\vec E_2}.d\vec{s}+\int_s{\vec E_3}.d\vec{s}+. . .=0$   Or is it possible that $\int_s{\vec E}.d\vec{s}=0$ does not imply $\vec E = 0$? We know that $12^2 = 144$ and that $38^2 = 1444$. Are there any other perfect squares in the form of $\frac{13}{9} (10^n - 1) + 1$ (i.e. $1$ followed by $n$ $4$'s), and how would we prove it? Searching through ODT documents without opening them? Let's say I have 2000 .ODT files. Their names are random numbers. How do I go about searching for, let's say, "pricing list"? How do I find the document I need without opening all of them one by one and checking to see if it's the right one?  Is there some program to search through the words in the documents without opening them? Why would parametric statistics ever be preferred over nonparametric? Can someone explain to me why would anyone choose a parametric over a nonparametric statistical method for hypothesis testing or regression analysis?   In my mind, it's like going for rafting and choosing a non-water resistant watch, because you may not get it wet. Why not use the tool that works on every occasion? Using Windows 10, I can't run certain commands from PowerShell but they all work correctly using cmd. PowerShell: node --version: node : The term 'node' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again. At line:1 char:1 + node --version + ~~~~     + CategoryInfo          : ObjectNotFound: (node:String) [], CommandNotFoundException     + FullyQualifiedErrorId : CommandNotFoundException  node.exe --version: no output. node.exe: opens a new black console (node.exe application itself). Get-Host | Select-Object Version: Version ------- 5.1.19041.906  Get-Command node: The term 'node' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the spelling of the name, or if a path was included, verify that the path is correct and try again. At line:1 char:1 + Get-Command node + ~~~~~~~~~~~~~~~~ + CategoryInfo : ObjectNotFound: (node:String) [Get-Command], CommandNotFoundException + FullyQualifiedErrorId : CommandNotFoundException,Microsoft.PowerShell.Commands.GetCommandCommand Get-Command node.exe: CommandType Name Version Source ----------- ---- ------- ------ Application node.exe 14.16.1.0 C:\Program Files\nodejs\node.exe $env:Path output has nodejs &amp; git &amp; other paths. cmd: everything works fine with any command. I have git and nodejs in my PATH env variable. What's wrong with PowerShell, what a strange behavior is this and how to fix it? Why can't variable names start with numbers? Can we make it more plain to recruiters that we're not open to relocation? The preference for being open to relocation is a checkbox, and if you don't check it, nothing about relocation shows up in your profile.  Evidently recruiters can't understand that a lack of "I am open to relocation" is the same thing as "I am not open to relocation", and so every single contact I've gotten through Careers is for a job that I would have to relocate for.  Can we make it more obvious to the recruiters (especially ones from Amazon) that we have not checked that box on purpose? Can't specify the 'async' modifier on the 'Main' method of a console app