i need help to demonstrate the next proposition using the definition you are giving me:   A space $(X, d)$ is chained if for all $\epsilon&gt; 0$ and each pair of points $x, y \in X$ there exist $x_{0}, ...., x_{n} \in X$ such that $x_{0}= x, x_{n}= y$ and $d (x_{i}, x_{i + 1}) &lt;\epsilon$ for each $i &lt;n$. Show that if $X$ is connected, then $X$ is chained.  I have a clear idea of how the demonstration is and i see it drawing pictures, but i don’t know how to write it, i would appreciate your help very much please
Show a connected metric space (X,d) is $\epsilon$-chainable for $\epsilon &gt;0$       $\epsilon$-chainable Definition      (X,d) is $\epsilon$-chainable if given any two points $a,b \in X$, $\exists$ a finite set $\{ x_0,x_1,...,x_n\}$ of points in X with $x_0 = a$, $x_n=b$ and $d(x_{n-1},x_n)&lt;\epsilon$, $\forall$ n      Clopen Definition      A set is clopen $iff$ its boundary is empty      Connected Metric Definition      Let $(X,d)$ be a metric space. It is connected $iff$ it has no proper subsets which are clopen      The following are equivalent:         $X$ is connected   The only subsets of $X$ that are clopen are $\emptyset$ and $X$   $\nexists$ non-empty, disjoint, open sets $A$ and $B$ such that $X=A\bigcup B$   $\nexists$ non-empty, disjoint, closed sets $A$ and $B$ such that $X=A\bigcup B$         Theorem      Let $(X,d)$ be a metric space, and let $S$ be any subset of $X$. Then $S$ is closed $iff$ $S^c$ is open.     So how would someone show this exactly?  Let the set $A$ be the set of all elements in $(X,d)$ that can be reached from $a\in X$ in a finite number of steps with length less than $\epsilon$.  Does this mean that there should be no isolated points? (an isolated point would be at distance infinity from any other point, or distance zero from itself, since $\epsilon &gt;0$ we don't want this?)  Knowing that $X$ is connected $\implies$the only subsets of X that are clopen are $\emptyset$ and $X$  Also, by construction of A;   A is the set of all $\epsilon$-chains between any element and $a$  And, so $A^c = \emptyset$ if $A$ was $\epsilon$-chainable (because A has to have a path from all elements to a, for this to be true, the complement wouldn't have anything in it)  Leaving only $A$ as the remaining clopen set  Does this make any sense? Or should an alternative be used, and how to formalize it.  Thanks.
How could a statement be true without proof? Godel`s incompleteness theorem states that there may exist true statements which have no proofs in a formal system of particular axioms. Here I have two questions; 1) How can we say that a statement is true without a proof? 2) What has the self reference to do with this? Godel sentence "G" can say that SUB(a,a, no prove) but could be this just arbitrarily judgement about non-provability of "a" because it may simply has a proof which is not yet revealed or discovered? Why does my iPhone power off when the battery is down to 30%? Issues Doing a String Comparison in LINQ For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution. For a prime $p \equiv 1$ or $3$  (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.   I'd like to see an elementary proof. Let $f : U \to V$ and $g : V \to W$ be linear transformations on the vector spaces $U$, $V$, and $W$. Supposedly,  $$ \dim(\ker(g \circ f)) = \dim(\ker(f)) + \dim(\ker(g) \cap \operatorname{im}(f)). $$  How might I go about proving that?    (Attempt:) The $\dim(\ker(g)\cap\operatorname{im}(f))$ term suggests to me that I should define a vector space $V' = \ker(g) + \operatorname{im}(f)$ to invoke the theorem that  $$ \dim(V') = \dim(\ker(g)) + \dim(\operatorname{im}(f)) - \dim(\ker(g) \cap \operatorname{im}(f)), $$  but I don't really see where to go from there. was created on Server Fault, answered on Server Fault, I upvoted some of those answers on Server Fault, and then moved by consensus to Super User.  Now that the question is on Super User, I can upvote the same answers I previously upvoted on Server Fault. 7 month old wants to sleep on its belly -- is it safe? How are connections differentiated from behind NAT routers? ICANN charges 18¢ per domain name registered regardless of its extension, but I pay $10 for a domain name per year with my registrar.   What do domain name registrars do for its customers to justify this extra $9.82?  Is there something that they do for our domain names other than registering it with ICANN?   I just read about the 7 people who hold the ICANN database key and got a little bit curious to know what the domain name servers do for us to justify this extra amount (12 times the actual fee)? Does the Great Old One warlock's telepathy from the Awakened Mind feature take an action? Does the Great Old One warlock's telepathy from the Awakened Mind feature take an action?  It doesn't seem too clear to our group and we have basically ruled it as equivalent to talking. Question about repeating decimal? For simple fraction, we can easily convert it to repeating decimal by calculator. Ex. $\frac 1 3 = 0.33333\ldots$, $\frac 1 7=0.(142857),\ldots$ But some fraction fraction like $10/29, 1/97,...$ The repeating part of them are too long, so it can't fully show on the calculator. So is there any algorithm  to find the repeating part for that fraction? Should we put the references in slides/poster when we already have them in paper? After reading this question about  and asking myself two questions about  and , I realize that these activities are for showing what you found, make the audience be attracted. If they need further explanation, they should read your paper, which has successfully defensed and proved that your work is validated. One important thing when you attempt to attract people is concentrate on the important, which means unnecessary things need to be trimmed. I think the reference is one of those. So, should we put the references in slides/poster when we already have them in paper? I really miss the graphical "Services" tool from older versions of Ubuntu, but I understand why it had to go. I always have terminals open anyway, so I use (because I have to now, I guess) service --status-all to see what services are running. This would be ok except that the command produces output like the following:   [ + ]  winbind  [ ? ]  wpa-ifupdown  [ - ]  x11-common   I'm guessing that those symbols mean something like listening, blocked, or stopped, but which is which? More importantly, why doesn't the man page say? Is there a way to get the raw SQL string executed when calling PDOStatement::execute() on a prepared statement? For debugging purposes this would be extremely useful. We have a view for a type of node that includes a Private File.  We cannot see a way to add an exposed filter, such as "Contains any word", and have it filter based on the file's name.  Are we missing something obvious? I moved postgresql's data directory by following the following steps:   Stop postgres cp -a source_data_directory destination_data_directory export PGDATA=destination_data_directory Changing data directory to destination_data_directory within postgresql.conf pg_ctl start   But when I run: "pg_ctl start" I get the following error. Can someone please help me figure out the error.   server starting postgres@Server:/usr/lib/postgresql/9.1/bin$ postgres cannot access the server configuration file "/home/destination_data_directory/postgresql.conf": No such file or directory I'm downloading some images from a service that doesn't always include a content-type and doesn't provide an extension for the file I'm downloading (ugh, don't ask).  What's the best way to determine the image format in .NET?  The application that is reading these downloaded images needs to have a proper file extension or all hell breaks loose. Should my first trip be to the country which issued my Schengen Visa? Einstein equation  where $E$ is energy, $m$ mass, and $c$ the speed of light in vacuum. So does it mean objects without any mass does not posses energy for eg lights photons does not have mass but how can they posses energy What is the command that lets me see what and where devices are mounted?  I'm having trouble changing songs on my old iPod, and I have a feeling it's because of the mount point.