How do i show that if every continuous function on $X$ is bounded, then $X$ is compact?
If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
Do you get Hex Warrior's Cha bonus with a two-handed Pact Weapon? Hex Warrior says:     you can touch one weapon with which you are proficient and that lacks the two-handed quality. When you attack with that weapon you can use your Charisma for attack and damage rolls.   later the ability says:     This benefit extends to every weapon conjured with the Pact of Blades feature, no matter the type.   (emphasis mine)  Does this allow me to use my charisma with 2 handed weapons(provided it is my Pact Weapon and my Hex Warrior Blade)?  This question is not the same as . That one is asking how many weapons, I'm asking if type means 2 handed or something else(Slashing or Ranged for examples). Regression Proof that the point of averages (x,y) lies on the estimated regression line How do you show that the point of averages (x,y) lies on the estimated regression line? When I log in with LightDM on my laptop running Debian Unstable, it recently started to hang for around 2 minutes until journalctl shows the message kernel: random: crng init done. When I press random keys on my keyboard while it hangs, it logs in faster (around 10 seconds). Before I didn't have this issue, is there any way I can fix it?  Edit: using linux-image-4.15.0-3-amd64 instead of linux-image-4.16.0-1-amd64 works, but I don't want to use an older kernel. I know references are syntactic sugar, so code is easier to read and write. But what is the difference between a pointer variable and a reference variable? Show $\text{Hom}_{Ab}(A,\mathbb{Q}) = \{0\}$ for finite $A$ Short-cut for switching to a given directory In the Ubuntu terminal I find myself spending considerable time to switch to a particular directory every time the computer starts up. Is there some way I can make this process easier? Is there some hotkey or some quick directory change that I can use (like speed dial on a phone)?   eg:     cd 1:Changes to saved directory one I have a question about something I'm wondering about. I've read somewhere that L'Hopitals rule can also be applied to complex functions, when they are analytic. So if have for instance:  $$ \lim_{z \rightarrow 0} \frac{\log(1+z)}{z} \stackrel{?}{=} \lim_{z \rightarrow 0} \frac{1}{(1+z)} = 1 $$  Now i'm wondering if this is correct? Also if we take $|z|&lt;1$, is it then correct?  Thanks, We've had a number of instances over on Stack Overflow involving users who get to the reputation level where they can comment, then spam comments across the site. Due to the lack of community review around comments, these spammed comments can remain on the site for months before someone notices and flags one. Moderators then have to go and manually remove each of these comments. In the last few months, I've personally had to deal with two users who spammed their website across 200+ questions each.  This isn't just limited to people spamming comments with a URL in them. We've had a number of users spray the same nonsense comment or insult across tons of posts and have no one seem to notice for months.  What I'd like is some kind of system-generated notification when a user has posted a barrage of copied-and-pasted comments. This could possibly be triggered when a user has posted a set number of identical comments in a row (4 of them, maybe) or maybe even if they've posted the same comment multiple times. We receive similar system-generated flags when users post duplicate answers, edit many of their posts in a row, or delete many of their own posts at a time.  This system-generated flag should ignore comments generated by review (which have the same text in them). I'm thinking that it shouldn't ignore comments generated by custom off-topic reasons, because someone using the exact same close reason over and over again might be something we want to look into.  The flag itself won't do anything to a user, and would only appear to moderators. I imagine there will be a number of false positives, but it should be easy for us to check these and identify the well-intentioned stock comments from the problematic ones.   We currently have no means of detecting comment spam, so even if this didn't have a great hit rate it would still be beneficial on the balance. Running sample queries for this seems to indicate that it wouldn't have that much noise.  This would allow us to catch comment spam early, and prevent it from living on the site or developing into a more serious problem. What does "What are you up to?" mean? I overheard my manager asking "What are you up to?"  What does that idiom mean? Is it an informal/negative way of asking?? Does driving an microcontroller's GPIO when it is powered off weaken the chip? We are using the  in a project. The way our design is now, the IC's GPIO pins will sometimes be driven high (to 3.3V) or low when it is powered-off. Will this weaken the chip? The absolute maximum ratings state the voltage on any digital I/O pin is -0.3V to VSupply+0.3V. If the answer is no, could the IC be weakened during power up?  The LTC5800-IPM does not have the most detailed datasheet. If it is not clear for that device in particular, I'd be interested to know for CMOS microcontroller GPIOs in general. Significance of the second focus in elliptical orbits Is there a way to make IE8 honour opacity on an `:before` pseudo element? Meaning of Simplify's tolerance parameter The standard parameters of Douglas-Peucker's simplify algorithm are geometry and tolerance (e.g  in PostGIS). What's the meaning of the tolerance parameter? I know that the bigger the value, the coarser the geometry will be. But does the number has any unit or it is just arbitrary? Whenever I install vsftpd on centos, I only setup the jail environment for the users and rest is default configuration of vsftpd. I create user and try to connect with filezila ftp client, but I could not connect with passive mode. I always change the transfer settings to active mode to successfully connect to the ftp server otherwise I get        Error: Failed to retrieve directory listing   So is there a way to change any directive in vsftp.conf file and we can connect with passive mode to the server? Help ID this plant, quite possibly a weed I found this plant near my South Indian home in , India. Most probably it could be some sort of weed. Anyhoo, is this plant of any medical or commercial value? Is it a remedy for something?   It has a white tap root with little veins off the main root It has a really small yellow flower It's leaves are similar to that of a clover, except for that it has three to five leaves        Merci d'avance!  (P.S. Forgive me if this is some very common plant that I should know about, I'm in 7th grade.) so I am trying to show that $f(x)$ divides $g(x)$ for all polynomials $g(x)$ satisfying that $g(A)=0$ where $f(x)$ is the minimal polynomial of a square matrix $A$.  I know from my professor that $f(x)$ is the minimal polynomial of $A$ so $f(A)=0$ so then $g(A)=0$. Therefore, $f(x)$ divides $g(x)$, i.e. $g(x)=f(x)h(x)$ where $h(x)$ is some other polynomial.  My prof told us to solve this question we need to let $b(x)=gcd((f(x),g(x))$ and by using Bezout's Identity, we need to show that $b(A)=0$. I am just having a little trouble trying to prove this. I know that this identity says:     For two polynomials $f_1(x)$ and $f_2(x)$, there exist polynomials   $g_1(x)$ and $g_2(x)$ such that $g_1(x)f_1(x) + g_2(x)f_2(x) = &gt; gcd(f_1(x), f_2(x))$.   And the uniqueness of minimal polynomial theorem is:     For a square matrix $A$, if $f_1(A) = f_2(A) = 0$ for two polynomials   $f_1(x)$ and $f_2(x)$, then $g(A) = 0$ for $g(x) = gcd(f_1(x), &gt; f_2(x))$. So the minimal polynomial of $A$ is unique up to a scalar.   So since $f(x)$ is already the minimal polynomial, I know that the $deg(b(x))=deg(f(x))$. I know how to prove the Bezout identity using the Euclidean algorithm, I'm just not exactly sure how I would show all of this using the notation he gave us of $b(x)=gcd((f(x),g(x))$. Any help at all would be appreciated. Prove that if $P$ is a $p$-Sylow subgroup of $G$ and $N \triangleleft\&gt; G$ then:    $ PN/N $ is a $p$-Sylow subgroup of $G/N$ $P \cap N $ is a $p$-Sylow subgroup of $N$ Email SMTP validator I need to send hundreds of newsletters, but would like to check first if email exists on server. It's called  validation, at least I think so, based on my research on Internet.    There's several libraries that can do that, and also a page with open-source code in  (), but I have hard time reading ASP Classic, and it seems that it uses some third-party library...    Is there some code for SMTP validation in C#, and/or general explanation of how it works? We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this:  Take $(\sqrt{2})^{\sqrt{2}}$  If it's rational, then you proved it, if it's irrational, take $((\sqrt{2})^{\sqrt{2}} ){^\sqrt{2}} = 2$ and you've proved it.  I'm wondering if you can raise $\pi$ or $e$ to a certain non-trivial real power to make it rational? And if not, where is the proof that it can't be done?   p.s. - I almost left out the real part, but then I realized that $e^{i\pi} = -1$. Not sure how to make tables, but:  For a binary operation $*$, and set $\{e,a,b,c\}$, in the Cayley table, $a*a$ can be filled with either the identity or an element different from both $e$ and $a$. If in the table the place for $a*a$ is filled with $e$, then the rest of the table can be filled out in two different ways.  What are the two different ways?  Thanks