Let $(X,d)$ be a metric space. This space is compact if any sequence $x_n \subset X$ has a convergent subsequence.   This is how I'm given the definition of a compact metric space and it confuses me. How come a definition is not an "if and only if" statement and instead an "if" statement. This seems more of a theorem to me and seems like there could be some metric spaces which are compact but do not have a convergent subsequence.  Also, if say, $E \subset X$ is compact,then would the values of those convergent subsequences be a member of $E$?  Thanks
In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions means "iff".  For example I am reading "Essential topology" and the following definition is written:     In a topological space $T$, a collection $B$ of open subsets of $T$ is said   to form a basis for the topology on $T$ if every open subset of $T$ can be written as a union of sets in $B$.   Should I assume the converse in such a case? Should I assume that given a basis $B$ for a topological space, every open set can be written as a union of sets in $B$?  This is just an example, I am not asking specifically about this sentence.
Set a global policy for floats positioning I have a document with many floats (figures, specifically).  I want to be able to change the global floats positioning policy from htbp to just h (or even H with the floats package).  For my-own reasons, I don't want to do this by appending the [h] argument to each float env.  One option that I can think of is using a macro:  \newcommand{\mypos}{h} % or htbp \begin{figure}[\mypos]...   I also prefer not to do that - It is cumbersome.  Is there a way to set the global float positioning argument globally? Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime. d'Alembert-like functional equation: $f(x+y)+g(x-y)=\lambda f(x)g(y)$ What's recommended for the number of users to use in qualitative and quantitative testing? How would one denote the set of all subsets of $A$ which have size $2$?  Would  $$\binom{A}{2}$$  be a good idea? Looking for book: contact with aliens, five different races, mismatched claws, biological imperative to breed Show $f$ is analytic if $f^8$ is analytic This is from Gamelin's book on Complex Analysis.  Problem: Show that if $f(z)$ is continuous on a domain $D$, and if $f(z)^8$ is analytic on $D$, then $f(z)$ is analytic on $D$. (I assume the intention is that $D$ is nice, i.e. open and connected)  I'm not exactly sure how to approach this. I'm guessing it has something to do with zeroes of $f$. For example, around non-zeroes, $f$ is analytic since $z^{1/8}$ is nonzero in a neighborhood. Or something along those lines. The details are eluding me.  Any help would be appreciated! When should we use "can", "could", "will", "would"? Most people say I wish I could, I wish you would.    Can we use I wish I can, I wish you will?    I'd like to know what the main differences between the usage of can/will and could/would are when wishing.  Is it related to the subjunctive? Because one says “I wish I were”, not “I wish I am”, so are would and could the subjunctive forms of can and will?  Or is it something else entirely? Elementary central binomial coefficient estimates Grub not showing on startup for Windows 8.1 Ubuntu 13.10 Dual boot Ok, so I'm such a newbie to Ubuntu and I bought a Windows 8 pre-installed laptop last month.  I updated to Windows 8.1 and then I thought about installing Ubuntu as a dual boot so I could mess around and learn more about it.   So I followed a Youtube tutorial (  ) and I got my stuff working fine.   The first few times I booted I got the GRUB menu instead of my default HP Boot OS Manager, and I was able to select my OS.   So I went to sleep and the next day I turned on my computer and the GRUB menu did not show up. I tried several times and it didn't automatically show up.  In order for me to see the GRUB menu I had to turn on my PC and on start had to press ESC to pause startup and press F9 to get boot options. Then from there I had to pick from OS Boot, Ubuntu, Ubuntu (Yes there were two Ubuntus available) and a default EFI file thingy.   When I click the first Ubuntu I get the GRUB Menu (I was too scared to try the second in case I screwed my laptop up) and I can safely load Ubuntu from there and use it (although I do have to increase my brightness every time I load Ubuntu because it somehow reduces my brightness to complete darkness on boot)  So my problem here is why isn't my GRUB showing on boot, after it worked on the first day?   My laptop is a HP TouchSmart j-078CA.  EDIT 1: FYI I just switched to Ubuntu LiveCD to check my Installation and found out that one of my partition is flagged as msftres. Here is a screenshot of my GParted on a LiveCD: Connectivity, Path Connectivity and Differentiability I have two questions which pertain to differentiability, connectivity and path connectivity. Ocasionally, I will encounter an author who defines connectivity in the following way:   An open subset $U$ of $\mathbb{R}^n$ is said to be connected if and only if given two points $a$ and $b$ of $U$ there exists a differentiable mapping $\phi: \mathbb{R} \rightarrow U$ such that $\phi(0) = a$ and $\phi(1) = b$.   This particular example is from Edward's Advanced Calculus of Several Variables p 84. Now, this is obviously not the standard definition we learn from topology which has nothing to do with differentiability but rather whether there exists two nonempty open subsets that comprise a separation. It also seems to me that what the author is really defining what it means for a space to be "smoothly path connected", which of course implies connectivity and, it seems to me, considerably more.  My first question is: Is "smoothly path connected", as defined above actually equivalent to "connected", in the topological sense, in $\mathbb{R}^n$?  Next, in Vector Calculus by Baxandall and Liebeck on p 150 the authors state the existence of a continuous path $\alpha$ from the closed interval $[0,1]$ to an open subset $D$ of $\mathbb{R}^n$ where $\alpha (0) = a \in D$ and $\alpha(1) = b \in D$ guarantees the existence of a differentiable path with the same properties. This claim is stated without proof.   My second question is: Can someone provide a reference to a proof of the above claim or explain why it is so? Prove if $\operatorname{ord}(a)=m$, $\operatorname{ord}(b)=n$, and $\operatorname{gcd}(m,n)=1$, then $\operatorname{ord}(ab)=mn$.   I was reading this and was thinking how this proof would look like. I tried to do it and am not sure if this is correct. Here is what I did:  If $a$ and $b$ commute then $(ab)^{mn} = a^{mn} * b^{mn} = (a^m)^n * (b^n)^m = 1 * 1 = 1.$ So $ord(ab) | mn$.  Now, take $k = \operatorname{ord}(ab) = m^\prime * n^\prime$ where m' is relatively prime with $n$ and $n'$ is relatively prime with $m$. By the result above $m' | m$ and $n' | n$. Now we have $((ab)^k)^{m/m'} = 1$ since $(ab)^k=1$. But on the other hand, by the commutativity:  $$((ab)^k)^(m/m') =$$  $$((ab)^(m' * n'))^(m/m') =$$  $$ a^{n'm} * b^{n'm} = $$  $$(a^m)^{n'} * b^{n'm'}=$$  $$b^{n'm'} = 1$$  This implies that $n' * m'$ is divisible by $n$. But $m'$ is relatively prime with $n$, so we must have $n' = n.$ By symmetry, $m' = m$. So $ord(ab) = mn$.  Just to say this again, I want to prove if $a$ and $b$ commute and $m$ and $n$ are relatively prime then   $$ord(ab)=mn.$$   The comments are vague. I guess this must be the answer then and there probably is not another way to do this. We are asked the following:      let $f(x)$ be a continuous function on $[0, 1]$ such that   $$\int _ {0} ^ {1} \ f(y) dy = 1$$   prove that $f(x) = 1$ for some $x ∈ [0, 1]$   The question asks to take into account Rolle's Theorem and Mean Value Theorem.  I'm aware of the definitions, but I am really unsure how to answer this question and my tutors are not really responding till after easter.    Could someone please help? Why is there a "y" involved?  Thanks you How can I read my hard drive’s SMART status in Windows 7? How can I read the  S.M.A.R.T. state of my HDDs while using Windows 7, either automatically or manually? How to make nautilus use file-roller in 17.10 like in 17.04? In Ubuntu 17.10 ubuntu upgraded Nautilus (File Browser) to version 3.26.0. New nautilus is not using file-roller anymore, it has  to an integrated compression mechanism.     This makes it complicated to create encrypted archives and/or different types compression archives that file-roller does provide.  How to make nautilus use file-roller as before?  Related question: SQL to LINQ Tool Is there a tool out there which can convert SQL syntax to LINQ syntax?   I just want to rewrite basic queries with join, etc., to . It would save me a lot of time. When flagging, there are two special case options: spam and rude or abusive.   What is spam, and when should I flag content as such? What is considered offensive, rude or abusive content? How does the spam flag differ from the rude or abusive flag? What is the effect of these special flags? When are these flags removed? Make Blender usable on high resolution screen Can I choose to delay my turn in combat? In D&amp;D 4e there was an option to Delay your initiative:     Perform your actions as desired and adjust your initiative to your new   position in the order.   Does this rule to change initiative order still exist in 5th edition? I cannot find it in the PHB. How can I get rid of those little white dots? I am using Blender 2.70a.