Consider a sequence $\{x_n\}$ in $S$. Where $S$ is a metric subspace. Given that every convergent subsequence of $\{x_{k(n)}\}$ converges to the same point say $x$. Prove that if S is compact, show that $\{x_n\}$ converges to $x$. Is my answer correct?
Let $(x_n)$ be a sequence of real numbers.  Prove that if there exists $x$ such that every subsequence $(x_{n_k})$ of $(x_n)$ has a convergent (sub-)subsequence $(x_{n_{k_l}})$ to $x$, then the original sequence $(x_n)$ itself converges to $x$ .              Thanks for any help.
Why do we need autoencoders? Recently, I have been studying autoencoders. If I understood correctly, an autoencoder is a neural network where the input layer is identical to the output layer. So, the neural network tries to predict the output using the input as golden standard.  What is the usefulness of this model? What are the benefits of trying to reconstruct some output elements, making them as equal as possible to the input elements? Why should one use all this machinery to get to the same starting point? SQL MAX of multiple columns? Convert from MySQL datetime to another format with PHP I have a datetime column in MySQL.   How can I convert it to the display as mm/dd/yy H:M (AM/PM) using PHP? How to start a terminal with certain text already input on the command-line? Rather than rephrasing my question, let me describe to you the desired user-case:  I create a short shell-script to run command "gnome-terminal --someoptionflagname 'my text to be posted'", and execute this script.  Gnome-terminal pops up, with command-line prompt followed by my text.   ie: fields@mycomputer:/$ my text to be posted  Can this be done? Is there a realistic way of implementing a multi-threaded model in PHP whether truly, or just simulating it. Some time back it was suggested that you could force the operating system to load another instance of the PHP executable and handle other simultaneous processes.  The problem with this is that when the PHP code finished executing the PHP instance remains in memory because there is no way to kill it from within PHP.  So if you are simulating several threads you can imagine whats going to happen.  So I am still looking for a way multi-threading can be done or simulated effectively from within PHP. Any ideas? I would like a way to convert a document from HTML to LaTeX on a Windows platform. What is my best option? Integral $\int_0^\infty \frac{\arctan(x) dx}{x(1+x^2)}$? How would one evaluate the integral $$\int_0^\infty \frac{\arctan(x) \,dx}{x(1+x^2)}$$?  I was told it had a nice closed form and could have been solved with differentiation under the integral sign; however, I tried to set $$I(\alpha) = \int_0^\infty \frac{\arctan(\alpha x) \,dx}{x(1+x^2)}$$ and got nowhere (the resulting integral was very messy). Is there a much more clever substitution that can be used to tackle the integral? How can I prove that prove $n^5 - n$ is divisible by $30$? I took $n^5 - n$ and got $n(n-1)(n+1)(n^2+1)$ Now, $n(n-1)(n+1)$ is divisible by $6$. Next I need to show that $n(n-1)(n+1)(n^2+1)$ is divisible by $5$. My guess is using Fermat's little theorem but I don't know how. How should you deal with (deliberate) selective deafness? Direction arrows on a circle I would like to plot circle (or another curve) with positive direction (and show it wit arrows).  This plots the circle  \documentclass{article} \usepackage{tikz} \RequirePackage{pgfplots} \usetikzlibrary{shadows} \usetikzlibrary{shapes} \usetikzlibrary{decorations} \usepackage{verbatim} \usepackage[active,tightpage]{preview} \PreviewEnvironment{tikzpicture} \setlength\PreviewBorder{5pt} \begin{document} \begin{tikzpicture}     \begin{axis}[             xmin=-1.5,xmax=2.5,ymin=-1.5,ymax=1.5,             axis lines=center,             ticks=none,             unit vector ratio*=1 1 1,            ]                   \addplot [very thick, smooth, domain=(0:2*pi)] ({1.+cos(deg(x))},{sin(deg(x))});        \addplot[only marks,mark=*] coordinates{(1,0)};         \end{axis} \end{tikzpicture} \end{document}   And I want to add arrows like that    Would you advise me how to add such arrows? Hom is a left-exact functor Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution )  1)$$\int_0^\infty { e^{-x^2} \ln x }\,dx = -\tfrac14(\gamma+2 \ln 2) \sqrt{\pi} $$  2)$$\int_0^\infty { e^{-x} \ln^2 x }\,dx = \gamma^2 + \frac{\pi^2}{6} $$  3) $$\gamma = \int_0^1 \frac{1}{1+x} \sum_{n=1}^\infty x^{2^n-1} \, dx$$  and lastly,  4) $$\zeta(s) = \frac{e^{(\log(2\pi)-1-\gamma/2)s}}{2(s-1)\Gamma(1+s/2)} \prod_\rho \left(1 - \frac{s}{\rho} \right) e^{s/\rho}\!$$  I personally think the last is obtained from a simple use of the Weierstrass factorization theorem. I'm unsure as to what substitution is used.  $\gamma$ is the Euler-Mascheroni constant and $\zeta(s)$ is the Riemannian Zeta function.  Thanks in advance. What is the problem that gets worse after you try to solve it? Example sentence - This problem is a _____ which gets complicated every time you try to solve it.   Is there such a word that represents a problem which will become worse if/when one tries to solve it?  Note - After seeing the comments and answers I'd like to add that this is not a can of worms or Pandora's box because it doesn't create a new problem but just worsens.   Example: Magic leaves protecting a fruit by enclosing it. Every time a leaf is touched, all the leaves close together further, that is more tight than before. Does equality of characteristic polynomials guarantee equivalence of matrices? I have a qualifying exam coming up in a couple days and I am just trying to understand some pathological examples I have in my notes. I will list a similar problem which I know the solution to and then the question.  True or False:   let $A,B \in \mathbb{Q}^{n \times n }$  Suppose $xI-A$ and $xI-B$ are equivalent then $\det(xI-A) = \det(xI-B)$.     I wrote in my notes this is true because any two matrices are equivalent in $\mathbb{Q[x]}^{n \times n}$ if and only if they have the same invariant factors.  Since the characteristic polynomial is a product of the invariant factors it follows that $\det(xI-A) = \det(xI-B)$.        True or False:   let $A,B \in \mathbb{Q}^{n \times n }$. If $\det(xI-A) = \det(xI-B)$ (in $\mathbb{Q}[x]$) then $xI-A$ and $xI-B$ are equivalent  in $\mathbb{Q}[x]^{n \times n}$   I think this is false because of the fact that equality of determinants is not strong enough to guarantee all invariant factors are equal but I do not have an example. Why Was It 'Essential' That Voldemort Kill Harry Potter? ‘So the boy ... the boy must die?’ asked Snape, quite calmly.   ‘And Voldemort himself must do it, Severus. That is essential.’   Deathly Hallows - page 551 - UK Hardcover - chapter 33, The Prince's Tale   Dumbledore tells Snape it is 'essential' that Voldemort be the one to kill Harry, I'm assuming in order for the piece of Voldemort's soul in Harry to be properly killed as well.  But why?   Hermione destroyed the Hufflepuff cup; Ron destroyed the Slytherin locket; Neville killed Nagini; Harry destroyed the diadem and Tom Riddle's diary. So the Horcruxes were not immune to destruction at the hand of someone other than Voldemort. And as it ended up, Harry himself wasn't even a Horcrux, but rather just an unknowing host to a parasitic bit of Voldemort's soul.   Why was it 'essential' that Voldemort be the one to kill Harry in order for the piece of Voldemort's soul to die? How can I test for a player wearing specific armor? How can I test for a player that is wearing an iron helmet named "test1" and an iron chestplate named "test2" in Minecraft Version 1.12? I'm working on a 2nd order passive low pass filter, consisting of two passive low pass filters chained together.         &ndash; Schematic created using   Let \$ H(s) = H_1(s)H_2(s) \$ where \$ H_1(s) \$ and \$ H_2(s) \$ are the transfer functions for each separate filter stage.  Then \$ |H(s)| = |H_1(s)||H_2(s)| \$  Knowing the magnitude of a passive low pass filter,   $$|H(s)| = \dfrac1{\sqrt{ (\omega R_1C_1)^2 + 1} } \times \dfrac1{\sqrt{ (\omega R_2C_2)^2 + 1} } = \dfrac1{\sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)}} $$  Then trying to find the cutoff frequency:  $$ \left(\dfrac1{\sqrt{2}}\right)^2  = \dfrac1{\sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)}} $$ $$ 2  = \sqrt{((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1)} $$ $$ 4  = ((\omega R_1C_1)^2 + 1)((\omega R_2C_2)^2 + 1) $$ $$ 4  = (\omega R_1C_1)^2(\omega R_2C_2)^2 + (\omega R_1C_1)^2 + (\omega R_2C_2)^2 + 1 $$  And I'm stuck. Research on the web tells me \$ \omega_c = \dfrac1{\sqrt{R_1C_1R_2C_2}} \$, but I can't find why? Can anybody show my the derivation to find this? Suppose $z\neq -1$ is a complex number of norm 1m $(|z| =1)$. I know that $z= n + mi$, what is the most straightforward way of solving this problem? I was also given the following sketch for a possible approach. How do I make exponential equation with points?  Here are my points:  x = 0, y = 0 x = 1, y = 4.75000 x = 2, y = 9.26250 x = 3, y = 13.54937 x = 4, y = 17.62191   this is same as   x = 0, y = 0 x = 1, y = (0 + 5) * 0.95  x = 2, y = ((5 * 0.95) + 5) * 0.95 x = 3, y = (((5 * 0.95) + 5) * 0.95 + 5)) * 0.95 x = 4, y = ((((5 * 0.95) + 5) * 0.95 + 5)) * 0.95 + 5) ) 0.95   as x value increase by 1, y value is added by 5 and multiplied by 0.95 i tried to make an equation for this but i couldn't  also i tried to use natural exponential but the initial value is 0 so it didn't work  please help me Which year does Spider-Man: Homecoming take place in? The first scene of the movie is presumably set in the immediate aftermath of The Avengers. The movie then fast-forwards to '8 years later'.  But Spider-Man already appeared in Civil War as a high school student, which he still is in this movie. So what's the timeline here? If Avengers was set in 2012 and Civil War in 2016, how can Homecoming be in 2020?  Or am I wrong in assuming the canonical timelines of the other movies?