I found the next exercise and its respective solution while studying calculus.  Let $(x_n)\in\mathbb{R}$ be a sequence. Suppose that every subsequence $({x_n}_k)$ has a &quot;sub-subsequence&quot; $({{x_n}_k}_j)$ that converges to $0$. Show that $\displaystyle\lim_{n\to \infty} x_n=0$  Proof:  Suppose that $x_n \not\to 0$. Then, there is an $\varepsilon _0&gt;0$ such that, for every $N\in\mathbb{N}$ $$|x_n|\geq\varepsilon_0$$, for some $n\geq N$.   Then, it's possible to construct a subsquence $({x_n}_k)$ such that $|{x_n}_k|\geq\varepsilon_0&gt;0$ for every $k\in\mathbb{N}$.   This subsequence $({x_n}_k)$ cannot have any subsequence that converges to $0$ (contradiction).  My questions are: Why can we affirm that it's possible to construct a subsequence $({x_n}_k)$ such that $|{x_n}_k|\geq\varepsilon_0&gt;0$, for every natural number $k$? And why can we affirm that the subsequence $({x_n}_k)$ cannot have any subsequence that converges to $0$?
Let $(x_n)_{n\in \mathbb{N}}$ be a sequence of real numbers. A well known result is:     The sequence $(x_n)_{n\in \mathbb{N}}$ converges to $a\in \mathbb{R}$ if and only if every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ converges to $a\in \mathbb{R}$.   Now, I'm working on an exercise which essentialy asks to prove the following:     The sequence $(x_n)_{n\in \mathbb{N}}$ converges to $a\in \mathbb{R}$ if and only if for every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ there is a (sub)subsequence $(x_{n_{k_j}})_{j\in \mathbb{N}}$ which converges to $a$.   Now, one direction is trivial. If in fact $(x_n)_{n\in \mathbb{N}}$ converges to $a$, then by the result I've mentioned, every subsequence of $(x_{n})_{n\in \mathbb{N}}$ converges to $a$. Let then $(x_{n_k})_{k\in \mathbb{N}}$ be an arbitrary subsequence. In that case, we know that it is a subsequence of itself, and hence we've found a (sub)subsequence which converges to $a$.  The other way around seems quite complicated for me. The reason is as follows: I must assume that every subsequence $(x_{n_k})_{k\in \mathbb{N}}$ has a (sub)subsequence which converges to $a$ and the proof that the original one converges to $a$.  My problem is exactly that "a". If it had all (sub)subsequences converging to $a$, then by the result i've mentionted, the subsequence $(x_{n_k})_{k\in \mathbb{N}}$ would converge to $a$. In that case, we would have every subsequence converging to $a$ and the result would follow.  My idea to prove the claim was then:  First remember that a subsequence is the restriction of the original one $x: \mathbb{N}\to \mathbb{R}$ to an infinite subset $\mathbb{N}'\subset \mathbb{N}$. In that case a (sub)subsequence is the restriction of $\tilde{x} = x|\mathbb{N}'$ to an infinite subset $\mathbb{N}''\subset \mathbb{N}$. But since $\tilde{x}$ is the restriction of $x$ to $\mathbb{N}'$ and since $\mathbb{N}''\subset \mathbb{N}'$, the restriction of $\tilde{x}$ to $\mathbb{N}''$ is itself the restriction of $x$ to $\mathbb{N}''$.  In other words, every (sub)subsequence can be viewed as a subsequence of the original one. Considering this, by hypothesis every subsequence has a (sub)subsequence which converges to $a$. But by this argument, this means that we can replace every subsequence with a convergent one. And then we can use the result I stated.  Is my idea correct? Is this really the way to prove the claim? My only doubt is that I need every subsequence converging to $a$ to use the result. In this case I can't see how I can argue that this implies every subsequence converges to $a$.  How can I finish this proof?
How long does it usually take for published articles to show up on Google Scholar? The title pretty much says it all - what's the typical delay with which Google Scholar indexes newly published articles? (Note I'm not talking about citations, but about the article showing up at all.)  Edit: the article in question went for over six months after being published (in a well-regarded Springer journal with its own Wikipedia page) without appearing on Google Scholar, until it received a citation, at which point it was added to Google Scholar almost immediately. One more motivation to write papers that get cited. :)  Edit 2: Another article in an even more mainstream journal took a little over seven weeks. How to store lots of small text in a SQL Server database? I am a Swedish citizen traveling to Kyrgyzstan via Russia. Do I need a transit visa while waiting for my flight iin 24 hours? Unable to cover get set in apex test class I am unable to cover the following class get set method. Need help in creating test class  public class Aresult{      public class Parent {         public String type {get; set;}         public String id {get; set;}         public String sequence_id {get; set;}         public String etag {get; set;}         public String name {get; set;}     } }   Please suggest  @isTest  public class CirriusGetItemsResultTest {     public static void myUnitTest(){         Aresult.parent a = new Aresult.parent();         a.type='hello';         a.id='hello';         a.sequence_id ='hello';         a.etag ='hello';         a.name ='hello';     } } I obtained a licence key for Windows 8 through DreamSpark. Is there a way to download a Windows 8 ISO image like you could for Windows 7? I am on a non-Windows machine, so I cannot use the Secure Download Manager to download Windows 8 directly using DreamSpark. When building console applications that take parameters, you can use the arguments passed to Main(string[] args).   In the past I've simply indexed/looped that array and done a few regular expressions to extract the values. However, when the commands get more complicated, the parsing can get pretty ugly.    So I'm interested in:   Libraries that you use Patterns that you use   Assume the commands always adhere to common standards such as . Why does chmod succeed on a file when the user does not have write permission on parent directory? I am trying to understand file/dir permissions in Linux.  A user can list the files in a directory using   cd test ls -l    Even if the user issuing above commands does not have read, write or execute permission on any of the files inside the test directory, still he can list them because he/she has read permissions on the test directory.   Then why in the following scenario user B can change permissions of a file he owns but does not have write permissions of the parent directory?  User A, makes a test directory and gives other users ability to write in it:  mkdir test chmod o+w test   User B, creates a file in test folder.  cd test touch b.txt   User A removes write permission of others from the directory  chmod o-w test   User B, can successfully change permissions, even though permissions are part of directory and this user does not have write permission on the parent directory of the file he owns  chmod g-r b.txt   why does chmod not fail since the user cannot modify the directory which has the file information - permissions etc? Drift velocity in Drude model An increasing number of funding organizations require publications on the research that they fund to be open access, i.e. available to the public without having to subscribe to a journal or pay a fee. Does anybody know where I can find a list of journals/publishers that do not allow material that they publish to be published this way? PDF upload silently fails I dunno since when, but upload of PDF images fails, this hurts especially on tex.SE.  Could this be an effect of an updated /etc/ImageMagick-6/policy.xml on the imgur server?  At least on my Ubuntu 18.04 LTS machine, this file was modified two weeks ago. What size should the HMAC key be with SHA-256? I'm trying to generate a secret key to be used for HMAC SHA-256 signature processing. I've seen many sample of keys with variable length from 32 characters to 96 characters.  What is the ironclad rule for this key size? How do I install and use the latest version of GNOME? I followed the instructions for compiling GNOME Shell and it did compile with a few errors here and there but it didn't run. I installed GNOME Shell using sudo apt-get install gnome-shell but I got version 2.31.x, while the latest is 2.91.5 or something. So is there a way for me to install the latest build?   Is there some repository available for the latest build? What is the difference between class and instance attributes? Prove that, $f'(0) \ge -\sqrt{2}$ for a function $f$ satisfying some conditions on $(-1,1)$. Let $f:(-1,1)\to \mathbb{R}$ be a twice differentiable function such that, $f(0)=1$, $f'(x)≤0$, $f(x)≥0$ and $f''(x)≤f(x)$ for all $x≥0$. Prove that, $f'(0)≥-\sqrt2$   Progress: I was able to prove $f'(0)≥-2$. For this I have applied Cauchy MVT for $f'$ and $g$ on $[0,x]$ for any $x\in (0,1)$ where $g: [0,x]\to\mathbb{R}$ is defined by, $g(x)=\sqrt{x+1}$. But, couldn't find another approach to get $-\sqrt{2}$. find pair of numbers whose difference is an input value 'k' in an unsorted array Possible Duplicate:          Is it correct to start a sentence with did.     Did my work.   instead of      I did my work.   while answering my emails? I'm hoping someone could check my work. I'm trying to simplify the equation for $x$. $$(\frac{x}{a})^{3.2} + (\frac{y}{b})^{3.2} = 1$$ $$\frac{a}{b} = \frac{174.1}{86}$$ Thus I began by substituting $a$ with: $$a = \frac{174.1}{86}b$$ $$((\frac{86}{174.1})(\frac{x}{b}))^{3.2} + (\frac{y}{b})^{3.2} = 1$$ $$((\frac{86}{174.1})(\frac{x}{b}))^{3.2} = 1 - (\frac{y}{b})^{3.2}$$ $$(\frac{x}{b})^{3.2} = \frac{(1 - (\frac{y}{b})^{3.2})(174.1)^{3.2}}{(86)^{3.2}}$$ $$x^{3.2} = \frac{(1 - (\frac{y}{b})^{3.2})(174.1)^{3.2}(b)^{3.2}}{(86)^{3.2}}$$ $$x^{3.2} = \frac{(1 - (\frac{y}{b})^{3.2})((174.1)(b))^{3.2}}{(86)^{3.2}}$$ $$x^{\frac{16}{5}} = \frac{(1 - (\frac{y}{b})^{3.2})((174.1)(b))^{3.2}}{(86)^{3.2}}$$ $$x = [\frac{(1 - (\frac{y}{b})^{3.2})((174.1)(b))^{3.2}}{(86)^{3.2}}]^{\frac{5}{16}}$$ Is my work correct, or am I completely off the board? How do I calculate the challenge rating of monsters that feature abilities that do not directly do damage?  For example, my homebrew monster has the two following abilities:     The target takes a Constitution saving throw. On a failure, it drops to 0   hit points.   This one could be used on a character with 10 hit points, or 100, which creates drastically different damage results depending on character level.     All creatures within 60ft take a Wisdom saving throw. Each target that fails is stunned for 1 round.   This can be extremely dangerous if this monster is paired with monsters that can make use of the stun condition, but it is virtually useless if the monster is alone.  I'm sure there are infinitely more examples of this type of thing.  How do we calculate CR for monsters that do not rely on standard attacks? I found  post about how to do Crow's Foot ERM Diagrams with Tikz, but it looks strange because the entity is outside the frame. However, it worked.  Then I found  even nicer version and tried to combine it, but sadly I'm not able to find out why it doesn't work. I'm not able to print two entities side by side and connect them with a crow's foot wire. They always print over each other...  How can I have both the lines and the style?  \documentclass{article}  \usepackage{tikz} \usetikzlibrary{shapes.multipart} \usetikzlibrary{matrix} \usetikzlibrary{positioning} \usetikzlibrary{shadows} \usetikzlibrary{calc}  \tikzset{     entity/.code={         \tikzset{             rounded corners,                          name=#1,             inner sep=2pt,             every entity/.try,         }%         \def\entityname{#1}%     },     entity anchor/.style={matrix anchor=#1},     every entity/.style={             draw,     },     every property/.style={         inner xsep=0.20cm, inner ysep=0.075cm, anchor=west, text width=1.75in     } } \def\property#1{\node[name=\entityname-#1, every property/.try]{\propertysplit#1;};} \def\properties{\begingroup\catcode`\_=11\relax\processproperties} \def\processproperties#1{\endgroup%     \gdef\propertycode{}%     \foreach \p in {#1}{%     \expandafter\expandafter\expandafter\gdef\expandafter\expandafter\expandafter\propertycode%             \expandafter\expandafter\expandafter{\expandafter\propertycode\expandafter\property\expandafter{\p}\\}%     }%     \propertycode% } \def\propertysplit#1:#2;{#1:\hfill#2}  \def\entitynamenode{% \node[every entity name/.try] (\entityname-name) {\entityname}; \draw (\entityname-name.south west) -- (\entityname-name.south east); \\[1ex] } \tikzset{   every entity name/.style={every property/.try, align=center} } \begin{document} \begin{tikzpicture}[every node/.style={font=\ttfamily}, node distance=0.5in] \matrix (m1) [entity=Employee] {     \entitynamenode     \properties{         Username:,         Password :  varchar(255),                 Name :  varchar(255),         LastName :  varchar(255)     } }; \matrix (m2) [entity=Student] {     \entitynamenode     \properties{         Username:,         Password :  varchar(255),                 Name :  varchar(255),         LastName :  varchar(255)     } }; \draw [-&gt;] (m1) -- (m2); % &lt;-- it break's here... %\draw [one to omany] (Employee-Username:) to node[above]{label} (Student-Username:); \end{tikzpicture} \end{document}      Edit: this is a Crow's Foot Notation for visualizing IT Database structures: Should OK button be on left of Cancel button or vice versa?   Are there any studies suggesting either of the solutions?