Or can we prove if A is both open and closed in the real line then it is empty or the whole line? Assuming the standard topology.
In other words, are $\emptyset$ and $\mathbb{R}$ the only open and closed sets in $\mathbb{R}$? Why/Why not?  I tried by assuming a set is equal to its interior points and contains its limit points.  A bounded set will not do since stuff like $[1,4]$ and $\{5\}$ will not work, though that is not really proof. Help please?  Anyway, it must then be unbounded.  If $a$ is a real number then $(a,\infty)$, $(-\infty,a)$, $[a,\infty)$ and $(-\infty,a]$ don't seem to cut it so it must be $\mathbb{R}$.
A Banach space is reflexive if and only if its dual is reflexive How to show that a Banach space $X$ is reflexive if and only if its dual $X'$ is reflexive? There is a socket related function call in my code, that function is from another module thus out of my control, the problem is that it blocks for hours occasionally, which is totally unacceptable, How can I limit the function execution time from my code? I guess the solution must utilize another thread. How to configure idle period for auto locking screen I just installed a fresh Ubuntu 12.04 (Precise Pangolin).   I noticed that, there is no screensaver settings, which is fine. I like to control the period of time after which the blank screen is shown,which automatically locks the screen.  I do not want to enable screensaver. I only wish to be able to change the idle time period required to lock the screen automatically. It seems to me, it is by default around 10-15 minutes. I currently have an Ubuntu-Windows Dual boot. However, I would like to try out a new OS so I burned Linux Mint to a USB stick.   I would like to install Linux Mint and remove Ubuntu while still keeping my windows dual boot. I've already backed up my Ubuntu setup so I can wipe it out.   In the Linux Mint installer, should I choosing the "install alongside windows" option do this? I use dd in its simplest form to clone a hard drive:  dd if=INPUT of=OUTPUT   However, I read in the manpage that dd knows a blocksize parameter. Is there an optimal value for the blocksize parameter that will speed up the cloning procedure? If $\sum_{n=1}^\infty \frac{1}{a_n}$ converges, must $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ converge? Suppose $\sum_{n=1}^\infty \frac{1}{a_n} = A$ is summable, with $a_n &gt; 0,$ $n = 1,2,3,\cdots.$ How can we prove that $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ is also summable?  This question came from a problem-solving seminar, but I'm quite stuck without a push in the right direction. I tried a few things, including Cauchy-Schwarz (which says $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n} &lt; \sum_{n=1}^\infty \frac{A}{n}$) and also the idea of assuming the latter series diverges and attempting to deduce the divergence of the former series from that, using facts such as $\sum a_n = \infty \implies \sum \frac{a_n}{a_1 + \cdots + a_n} = \infty$. Nothing has worked so far. I need to calculate no of possible substrings containing "00" as a substring. I know the length of the binary string.  Eg: for a string of length 4, possible substrings are: 0000 0001 0010 0011 0100 1000 1001 1100  I just need the number of possible combinations, not enumerate all of them. The computer gave me this output in a window:  E: Encountered a section with no Package: header E: Problem with MergeList /var/lib/apt/lists/archive.ubuntu.com_ubuntu_dists_natty_main_binary-i386_Packages E: The package lists or status file could not be parsed or opened.   How can I fix this? How can I get a copy of a Canadian visa refusal letter? I have applied for a Canadian Temporary Resident Visa (TRV); a previous paper application was refused in 2016. I have lost the copy of my refusal letter and want to submit it when I apply for my new visa. How how can I get the a copy of the 2016 refusal letter from Embassy? LuaLaTeX protrusion problem with Negative value Let $(M,d)$ be a metric space, and $A\subset M$. By definition, $A$ is said to be compact if every open cover of $A$ contains a finite subcover.  What is wrong with saying that, in $\mathbb{R}$, if $I=(0,1)$, we can choose $G=\{(0,\frac{3}{4}), (\frac{1}{4}, 1)\}$, which satisfies $I \subset \bigcup_{U\in G} U$, but we can't extract a finite subcover, so $I$ is not compact. Is $G$ a finite subcover of $G$, so it is not a valid cover for proving this? I would take $\cup_{n\in\mathbb{N}} (\frac{1}{n},1)$ in order to prove this, can we conclude that every open cover is necessarily a infinite union of open sets $\neq \emptyset$? Calculate first difference by group in R I'm playing around with height maps (bitmaps), trying to create some of my own in my game, and for that I need to implement some basic drawing methods. I've quickly realized that drawing straight lines isn't as basic as I thought.  It simple's if your points share an X or Y coordinate, or if they are aligned so you can draw a perfectly diagonal line. But in all other cases its trickier.  What algorithm do you use to determine what pixels need to be coloured for it to become a "straight" line ? Will new Minecraft features appear in existing worlds? I'm a new Minecrafter, and was wondering if new updates are backwards compatible. Specifically, will the new "Adventure Update" features (Beta 1.8) appear in existing worlds created with Beta 1.7 and earlier? Or will I need to create a new world for this version? I have scripts that expect specific drive letters for hard drives. Windows sometimes assigns the different drive letters (e.g. I expected my Maxtor drive to be E:, but it's now F:, therefore my scripts will fail), depending on the order of how the drives are plugged into my laptop.  In Windows 7, how do I assign a permanent drive letter to a drive? Solving linear congruence (modular inverse or fraction) via gcd Bezout equation Isn't finding the inverse of $a$, that is, $a'$ in $aa'\equiv1\pmod{m}$ equivalent to solving the diophantine equation $aa'-mb=1$, where the unknowns are $a'$ and $b$? I have seem some answers on this site (where the extended Euclidean Algorithm is mentioned mainly) as well as looked up some books but there is no mention of this. Am I going wrong somewhere or is this a correct method of finding modular inverses? Also can't we find the Bézout's coefficients by solving the corresponding diophantine equation instead of using the extended Euclidean Algorithm? Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$ I am stuck on proofs with subsequences. I do not really have a strategy or starting point with subsequences.   NOTE: subsequential limits are limits of subsequences  Prove: $a_n$ is bounded $\implies \liminf a_n \leq \limsup a_n$   Proof:  Let $a_n$ be a bounded sequence. That is, $\forall_n(a_n \leq A)$.  If $a_n$ converges then $\liminf a_n = \lim a_n = \limsup a_n$ and we are done.  Otherwise $a_n$ has a set of subsequential limits we need to show $\liminf a_n \leq \limsup a_n$:  This is where I am stuck... Proving that $n$-cycle has order $n$ Currently, the number of "votes cast" excludes votes on deleted content. Once something gets deleted that you voted on, that vote vanishes from your stats. This of course happens frequently especially with down-votes - if you're a good voter, stuff you downvote often ends up getting deleted.  Seeing as the idea behind the public display of total votes you've cast is to encourage voting, is this really fair?   Also, doesn't it distort greatly all voting stats, especially when looking at candidates for moderator elections and such? A 398MB directory was only compressed to 393MB using 7Z and Normal ZIP compression. Is this normal? If so, why do people continue to use ZIP on Windows?