Is $ d(U,V) = 0 $? True/false Given   $(X, d)$ be a metric space and let $U$ and $V$ be non-empty subsets of $X$. Define   $d(U,V) = \inf\{d(u,v) : u \in U, v\in V\}$.   Is the following statement is true/false     If $U$ and $V$ are compact and disjoint, then $d(U,V) &gt; 0$   My attempt : I thinks  this statment is false  I take  $U= [-1,0]$ and $V= [1,2]$ both $U$ and $V$ are compact and disjoint .  Now for   each $\epsilon &gt;0$ , we have  $\frac{-\epsilon}{2} \in [-1,0] $and $\frac{\epsilon}{2} \in [1,2]$ so that   $d([-1,0], [1,2] \le d(-\frac{\epsilon}{2}, \frac{\epsilon}{2}) = \epsilon $  since $d([-1,0], [1,2])\le \epsilon $ for  every $\epsilon &gt;0$. Now  we conclude that  it is $0$ so its  contradicts that  $d(U,V) &gt;0$  Is its  true ?
Given a compact set $K$ and closed set $E,$ prove: $E\cap K\neq \emptyset $ Let $K\subseteq \mathbb{R}^n$ be a compact set and $E\subseteq \mathbb{R}^n$ a closed set, such that: $$\inf\{d(x,y):x\in K, y\in E\}=0$$ where $d(x,y)=\sqrt{\sum_j(x_j-y_j)^2}$ Prove: $E\cap K\neq \emptyset $  I tried to work with the definitions of compact and closed sets, and I understand that from what's given of the distance between elements in $K$ and $E$, that they are very close,  but couldn't really get further. Any help appreciated.
Show that $U_n:=\left(1+\dfrac{1}{n}\right)^n$, $n\in\Bbb N$, defines a monotonically increasing sequence.   I must show that $U_{n+1}-U_n\geq0$, i.e. $$\left(1+\dfrac{1}{n+1}\right)^{n+1}-\left(1+\dfrac{1}{n}\right)^n\geq0.$$  I am trying to go ahead of this step. I have a problem with a virus in windows 7, I don't know what kind of virus is infected my OS, I installed kasersky antivirus, and clear all detected viruses. After that I can't open any antivirus sites, can't ping any antivirus sites. example:  C:\Windows\system32&gt;ping kaspersky.com Pinging kaspersky.com [38.117.98.231] with 32 bytes of data: General failure. General failure. General failure. General failure.  Ping statistics for 38.117.98.231:     Packets: Sent = 4, Received = 0, Lost = 4 (100% loss),   When I ping to other sites, I get reply normally: example woth pinging to google.com  C:\Windows\system32&gt;ping google.com  Pinging google.com [31.7.160.224] with 32 bytes of data: Reply from 31.7.160.224: bytes=32 time=2ms TTL=61 Reply from 31.7.160.224: bytes=32 time=2ms TTL=61 Reply from 31.7.160.224: bytes=32 time=1ms TTL=61 Reply from 31.7.160.224: bytes=32 time=1ms TTL=61  Ping statistics for 31.7.160.224:     Packets: Sent = 4, Received = 4, Lost = 0 (0% loss), Approximate round trip times in milli-seconds:     Minimum = 1ms, Maximum = 2ms, Average = 1ms   I tryed any antispyware/antimalware softwares, but null results.  Anyone else encountered this problem? How to remove duplicate entries in dash? Why is voltage added when batteries are connected in series but not in parallel? As I understand, the different voltage between two object is caused by the total different charged between two object. Thus, the higher extra electron between two opposite charged object, the higher voltage between them, isn't it? So if i have two battery or capacitor connect in parallel, the negative connect to negative, the positive connect to the positive. Caused the charge from each plate to be double. So that's as my expected, it must be double voltage, but ironically why it's out of my expected and the reality is vice-versa? Checking to see if a user has a licence for a managed package So I'm aware that there is a method on UserInfo called isCurrentUserLicensed which lets you check if the user running the code has a licence for a namespace that you specify, i.e. for a particular managed package.  What I want to know is whether there is a way I can test to see if some other user (assume I have their ID) has a licence for a particular package. Has anybody found a way to do this? What are instructions on how to download videos from YouTube?    Notice: Terms of Service Violation  Please be aware that by following any of the answers below, you will be violating YouTube's . In particular, from Section 5.B.:     Content is provided to you AS IS. You may access Content for your information and personal use solely as intended through the provided functionality of the Service and as permitted under these Terms of Service. You shall not download any Content unless you see a “download” or similar link displayed by YouTube on the Service for that Content. You shall not copy, reproduce, distribute, transmit, broadcast, display, sell, license, or otherwise exploit any Content for any other purposes without the prior written consent of YouTube or the respective licensors of the Content. YouTube and its licensors reserve all rights not expressly granted in and to the Service and the Content. Can I use lens brand X on interchangeable lens camera brand Y? I've got some lenses of brand X, but I've just got a new camera of brand Y. How can I tell if it's possible to use my lenses on my camera? I accept that I may lose a lot of automatic functionality like autofocus, aperture control and metering when using an adapter to connect the two. Enable SSO for Salesforce Marketing Cloud I need to enable Single Sign On for Salesforce Marketing Cloud account where Identity Provider will be a third party and Marketing Cloud will be the Service Provider. I already got SSO Metadata option enabled for my account. I however, need an Entity ID and Federation ID. I am not clear about how to carry the entire process. If anyone of you guys has done SSO enabling for Marketing Cloud, please share steps. In my experience, input type="text" onchange event usually occurs only after you leave (blur) the control.  Is there a way to force browser to trigger onchange every time textfield content changes? If not, what is the most elegant way to track this “manually”?  Using onkey* events is not reliable, since you can right-click the field and choose Paste, and this will change the field without any keyboard input.  Is setTimeout the only way?.. Ugly :-) Lightsaber blade "weight" in-universe What is $\gcd(0,a)$, where $a$ is a positive integer? I have tried $\gcd(0,8)$ in a lot of online  gcd (or hcf) calculators, but some say $\gcd(0,8)=0$, some other gives $\gcd(0,8)=8$ and some others give $\gcd(0,8)=1$. So really which one of these is correct and why there are different conventions? I know some countries such as UK has the steering wheels of their cars on the right side. But as we can see, most of the Europian countries uses it at the opposite side. Since I used to drive a car with left-steering wheel I found it impossible to drive a car with right-steer!  Not curiosity, but I am trying to map if this issue is related to User Experience or not. (mód 1) is not copied correctly Please consider the following MWE:  \documentclass{article} \usepackage[spanish]{babel} \selectlanguage{spanish} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document} Works: áéà  Does not work: \(\pmod1\) \end{document}   The áéà text is copied correctly, but if I try to copy (mód 1) it says: (m´od 1).  Does anyone know why this happens? How is it solved?  Thanks! Why does the beta distribution become U shaped when $\alpha$ and $\beta$ <1? In the Beta distribution (used to model Bernoulli probabilities), the $\alpha$ and $\beta$ parameters can be interpreted as the number of heads$+1$ and the number of tails$+1$ seen. So, if they were both $2$, it would lean towards the coin being fair and have a maximum at $0.5$. If they are both $20$, the distribution would become even surer we're dealing with a fair coin and peak even more at $p=0.5$.  What I don't get is its behavior when $\alpha$ and $\beta$ both become $&lt;1$.   In that case, it becomes U-shaped and the density peaks at $p=0$ and $p=1$. Meaning the coin is likely to be two-sided. I know there is an intuition for this since I think I had an idea about it a long time ago. However, I've been trying to recollect all day and can't piece it together. Does anyone have an intuition? If the laws of random variables weakly converge to a point mass at $c$, then the random variables converge in probability to $c$. This is question 10.3.1 from Rosenthal's First Look at Rigorous Probability.  Suppose $\mathcal{L}(X_n) \Rightarrow \delta_c$ for some $c \in \mathbb{R}$. Prove that $\{X_n\}$ converges to $c$ in probability.  Here $\mathcal{L}(X_n)$ refers to the distribution of $X_n$.  Is the following proof correct?  Is a simpler proof available?  Is there a simple proof using Skorohod's theorem?  I attempted to prove the contrapositive. Suppose $X_n \not \to c$ in probability.  This means that there is an $\varepsilon &gt; 0$ so that for all $N$, there exists an $m &gt; N$ such that  \begin{equation}   P(|X_m - c| \geq \varepsilon) &gt; \eta, \text{ for some $\eta &gt; 0$}. \end{equation}  Consider  \begin{equation}   B := \{x \in \mathbb{R} : |x - c| \leq \varepsilon\} \in \mathcal{B} \text{ the Borel sets on $\mathbb{R}$}. \end{equation}  Then $B^c \in \mathcal{B}$ and denoting the distribution of $X_n$ as $\mu_n$ we have  \begin{equation}   \mu_n(B^c) = P(X_n \in B^c) = P(|X_n - c| \geq \varepsilon) &gt; \eta. \end{equation}  For all $N$, there exists an $m &gt; N$ such that the cumulative distribution function of $X_m$ is not the same as the cumulative distribution function of $\delta_c$; in particular it's mass changes somewhere at least $\varepsilon$ distance from $c$. Finally because $\delta_c(\{x\}) = 0$ for all $x \neq c$ this completes the proof. Why is paper more frangible when it is wet? My four-year-old daughter asked me why paper tends to fall apart when wet, and I wasn't sure.  I speculated that the water lubricates the paper fibers so that they can untangle and separate more easily, but I really wasn't sure. Do I have an SSD and a hard drive on my MacBook Pro? I looked at the details provided in  my system report/storage and I noticed that it appears I have a hard drive of 499.05GB and an additional SSD drive of 499.42GB. Does this mean I have 1TB of data? I only have 139GB available so do I have 500GB 'hiding' somewhere? (If so where??)  I am unsure how to interpret this information- why does it list both storage types? Is it duplicating the information?     Macintosh HD:      Available:    139.8 GB (139,800,666,112 bytes)   Capacity: 499.05 GB (499,046,809,600 bytes)   Mount Point:  /   File System:  Journaled HFS+   Writable: Yes   Ignore Ownership: No   BSD Name: disk1   Volume UUID:  EF0908C8-XXXX-XXXX-XXXX-XXXX80333709         Logical Volume:  Revertible:  Yes (unlock and decryption required)   Encrypted:    Yes   Encryption Type:  AES-XTS   Locked:   No   LV UUID:  8CC9C88D-XXXX-XXXX-XXXX-XXXX2002329B        Logical Volume Group:   Name:    Macintosh HD   Size:    499.42 GB (499,418,034,176 bytes)   Free Space:  18.9 MB (18,903,040 bytes)    LVG  UUID: EE040070-XXXX-XXXX-XXXX-XXXXA83B309         Physical Volumes:      disk0s2:     #is this a second disk?      Device Name:  APPLE SSD SM0512G   Media Name: APPLE SSD  SM0512G   Media   Size:   499.42 GB (499,418,034,176 bytes)   Medium Type:  SSD    Protocol:  PCI    Internal:  Yes   Partition Map Type: GPT(GUID Partition Table)  #does this mean its partitioned?   Status: Online   S.M.A.R.T. Status:  Verified   PV UUID:    A1391405-XXXX-XXXX-XXXX-XXX2E8FCDF3 Evil thoughts without actions? Cross-reference packages: which to use, which conflict? How to remove white space before reference?