Let $K$ and $L$ be nonempty compact sets, and define $$d= \inf \{|x-y|: x \in K \wedge y \in L\}.$$   Show that it is possible to have $d = 0$ if we assume only that the disjoint sets $K$ and $L$ are closed (i.e. they are no longer bounded).   I was thinking that essentially this problem breaks down into where we can use the unboundedness of $K $ and $L$ in order to show that at some point $d=0$. In other words, I'm looking for an example in which this occurs so that I fulfill the requirement that "it is possible." I thought about K and L being the union of alternating bounded intervals that go on infinitely, Now, I'm simply getting lost in the proof the d can =0. I think I can find that d approaches 0. Is that sufficient?
Let $A$ and $B$ be two sets of real numbers. Define the distance from $A$ to $B$ by $$\rho (A,B) = \inf \{ |a-b| : a \in A, b \in B\} \;.$$ Give an example to show that the distance between two closed sets can be $0$ even if the two sets are disjoint.
I've had  on the web since 1999, and I'd like to add a page on why complex numbers are (or should be) considered to be numbers.  I'm frequently asked that question.  This is not a question about the existence or usefulness of complex numbers as they do exist as mathematical entities and they are very useful.  It is a question about what makes them numbers.  It could be broadened to ask what constitutes numbers, but I'm specifically interested in complex numbers.  Perhaps it's just historical and sociological, but I'm hoping for an answer that there's something intrinsic about complex numbers that makes them fit into the category of number. Reduce Spacing In $$ In math, one often needs to put a letter inside the symbols &lt;&gt;, e.g. to denote a cyclic group &lt;x&gt; generated by some element x. However, plain text displays the symbols &lt; and &gt; as an upside down exclamation point and an upside down question mark, respectively, while math type displays a large space like so:  &lt; x &gt;    Is there some way to reduce this spacing? Virtualbox Kernel driver not installed How do you create grass? Are there any clones/alternatives for running a Stack Exchange style Q&A site? How to pass an array of Uri between Activity using Bundle Should we have another site for frameworks (and alike) recommendations? There is . Is the site appropriate for stuff related to requesting frameworks and libraries? Or should we have another site for that (e.g. name it Frameworks Recommendations or stuff alike)?  I can ask stuff there like: I need a software satisfying W, X, Y, and Z, but I'd like -some day- ask like: I need a Python library satisfying W, X, Y, and Z. The audience may be (and perhaps will be) different.    Is that website on-topic for stuff like that? Or should we open another proposal for that?  No: It's not about asking for a giga-dump for subjectivish questions. The linked-to-dupe question was written and answered with an old mindset and without Software Recommendations in mind. Also it does not consider this specific kind of questions but all the crap in one place. So, no,  is not an appropriate dupe mark for this question. Jquery change image on rollover with fade A relative managed to change her password, yet can not input the correct password.  She does not want to reset the phone to avoid loss of data (pictures, contacts, messages, etc.).  Will the phone eventually stop accepting logon attempts? Add review queues to Android app Short story about survivors on Earth after the atmosphere has frozen The kernel and range of the powers of a self-adjoint operator Suppose $L: V \rightarrow V$ self-adjoint (on a finite-dimensional space $V$); $b$ is a real number. Show   The minimal polynomial has distinct roots; $\ker(L) = \ker(L^k)$ for $k\geq1$; $\operatorname{im}(L) = \operatorname{im}(L^k)$ for $k\ge 1$; $\ker(L-bI) = \ker((L-bI)^k)$ for $k\geq1$;     My thoughts:  1) I think I'm supposed to assume root is repeated and reach a contradiction but I'm not sure how to proceed mathematically with that.  2-4) pretty sure there's some huge theorem that I'm missing. Suggested edits in top-bar put flags over the winterbash snowflake Suggested edits in the top bar put the flag count right over the winterbash snowflake. Looks pretty ugly if you ask me... How to assign a SSL Certificate to IIS7 Site from Command Prompt Can you advise me whether it is possible or not to assign a SSL Certificate to a website in IIS7 using the APPCMD application?  I am familiar with the command to set the HTTPS Binding  appcmd set site /site.name:"A Site" /+bindings.[protocol='https',bindingInformation='*:443:www.mysite.com']   and how to obtain current mappings   %windir%\system32\inetsrv\Appcmd   but can not seem to find any way to map a site to a certificate (say the certificates hash for example) Removing phone number from iMessage I'd like to remove my phone number from my iMessage account, but still have my phone enabled to receive messages sent to my email address (also used on iPad, Mac). Is there a way of doing this? What is "all" in isset($wp_filter['all']) I'm a newbie in WordPress. I'm trying to understand how it generates the XHTML pages. I guess it's the function of do_action( 'hook-name' ) , but I'm stuck with the understanding of the instruction isset( $wp_filter['all] ) when reading the function do_action because I don't understand what is the meaning of the word all.  Is it a PHP feature or a WordPress one? C# 4.0 optional out/ref arguments Determine all of the accumulation points of the following sets in $\mathbb{R}^1$ and decide whether the sets are open or closed or neither.  The set $X$ of all numbers of the form $2^{-n} + 5^{-m}$ where $m,n = 1,2,\ldots$.   Claim: The set of accumulation points is the following: $$\left\{\frac{1}{2^n}: n \in \mathbb{N} \right\} \cup \left\{\frac{1}{5^n}: n \in \mathbb{N}\right\} \cup \{0\}.$$ Let $n \in \Bbb{N}$ and $r&gt;0$, and consider the open neighbourhood $B\left(\frac{1}{2^n},r\right) = \left(\frac{1}{2^n} - r,\frac{1}{2^n} + r\right)$. For the given $r$ we can find $m$ large enough such that $5^{-m} &lt; r \implies \frac{1}{2^n} + \frac{1}{5^m} &lt; \frac{1}{2^n} + r$, and $\frac{1}{2^n} - \frac{1}{5^m} &gt; \frac{1}{2^n} - r$; therefore $5^{-m} + 2^{-n} \in B\left(\frac{1}{2^n},r\right)$. A similar argument shows that each element of $\{5^{-m}: m \in \mathbb{N}\} \cup \{0\}$ is an accumulation point of $X$. Now consider $y \in X$. $B(y,r)$ isn't a subset of $X$ since for all $r &gt; 0$, $B(y,r) \subset (\mathbb{R} - \mathbb{Q}) \not\subset \Bbb{Q}$, hence $B(y,r) \not\subset X$. Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$. Prove that for every $a, b\in R\setminus\{0\}$ is correct this inequality:  $$\frac{|a+b|}{1+|a+b|}&lt;\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}.$$ Where does the damage of relocated dashes land? The end location of a dash or jump can be changed by an ability that pushes () or pulls () you. So for example I try to  on the adc, but a Blitzcrank pulls me, preventing me from landing on my chosen spot.  Will I deal my AoE damage around the adc, where I tried to land, or around the Blitzcrank, where I ended up landing?