A sequence is defined to be bounded provided there is a positive real number M such that |an|&lt; M for every natural number n. I need to prove that {an} is bounded if and only if there are real numbers a and b with a &lt; b such that {an} is a sequence in [a, b]. So far I have been unable to produce a formal proof. Please help.
Why is it that a sequence in a closed, bounded interval [a, b] is a bounded sequence? What is the formal proof of this proposition from the definition of a bounded sequence?
Radeon HD 2000, 3000, 4000 on Ubuntu 12.04.2 and higher: fglrx (legacy) unsupported, what to do? After upgrading to 12.10 quantal, the packaged version of fglrx no longer works. I discovered that this is because there is a separate 'legacy' fglrx driver for the HD 2k-4k series cards, but it is incompatible with the xorg server on 12.10.  This is the most current version of the driver for HD 2000 through HD 4000 series cards. You can't use the non-legacy fglrx driver, but you can use the open-source radeon driver if you prefer your WM compositing to be laggy and your YouTube videos to play like they would on a Pentium MMX series:    Usually this driver can be installed in the following way, necessary because apt-get install fglrx would pull in the non-legacy driver:  wget http://www2.ati.com/drivers/legacy/amd-driver-installer-12.6-legacy-x86.x86_64.zip unzip amd-driver-installer-* sudo sh ./amd-driver-installer-*.run --buildpkg Ubuntu/quantal sudo dpkg -i fglrx*.deb sudo aticonfig --initial -f   If you use a different version of fglrx (for example, a newer 12.9 that doesn't support those cards) then the final command will give you an error no supported hardware detected or something similar. However, everything works at this point and you will get a reasonable xorg.conf:  ... other stuff  Section "Device"         Identifier  "aticonfig-Device[0]-0"         Driver      "fglrx"         BusID       "PCI:1:5:0" EndSection  ... other stuff   At this point you're supposed to reboot and everything will be working with the fglrx driver. However, upon rebooting, you'll be treated to the following errors in Xorg.0.log when fglrx attempts to load:  (EE) Failed to load /usr/lib/xorg/modules/drivers/fglrx_drv.so: /usr/lib/xorg/modules/drivers/fglrx_drv.so: undefined symbol: noXFree86DRIExtension   Some searching around will show that this is a problem with the legacy ATI drivers not supporting xserver 1.13 or newer. () ATI has released a fixed driver for its most recent (HD 5000 series or later) cards, but not for the 'legacy' cards yet. The non-legacy ATI drivers can't be used with the old cards.  What should an Ubuntu user, using one of these HD 2000-4000 series cards, do?    Wait for an updated 'legacy' ATI driver that properly works with xserver 1.13?   Downgrade back to 12.04 Precise, which uses xserver 1.11? Try to downgrade xserver on 12.10 Quantal to 1.12, which could possibly break Unity and GNOME? Forced upgrade to HD 5000 series or later card? (Not possible with integrated graphics...) Some other 1337 action that fixes this problem painlessly? Symbolizing using RGB values from Excel in ArcGIS Desktop? I have an excel file with 2 attribute columns, 1-feature info 2-Number of color (RGB) that the current feature should be symbolize. This file file can be joined with the layer using the "feature info" field which is unique (join by attribute table). So i have as attribute the color number attribute for every polygon feature.  How can  i use this this information to symbolize every feature with their attributed color already imported from the excel sheet? I (think) that I've configured a static IP address in /etc/network/interfaces:  # The primary network interface  auto eth0  iface eth0 inet static     address 10.1.1.2     netmask 255.255.255.0     network 10.1.1.0     broadcast 10.1.1.255     gateway 10.1.1.1   And I change the file /etc/resolv.conf to include the desired nameservers.   Then I restart networking sudo /etc/init.d/networking restart  But after some hours resolv.conf always reverts back to the DHCP nameserver.   I should also say that this is Ubuntu Server with only l0 and eth0 interfaces. Difference between ' and " on command line (bash)? I used to use '' and "" (single and double-quotes) interchangeably on the command line, but I recently noticed that '$HOME/some/dir' is not expanded, while "$HOME/some/dir" is. I searched around a little bit and found that "" allows some protection of special characters, while '' allows full protection. So what other characters are protected by '' and not ""?  EDIT: What are practical situations in which these differences might be significant? ravel: opposite meanings? I just came over this syntax in some of the questions in this forum, but Google and any other searchengine tends to block out anything but letters and number in the search so it is impossible to search out "=>".  So can anyone tell me what it is and how it is used? Automatically Locate Included Images Do you think it's possible to include images into LaTeX without specifying the exact path to them. So instead of:  \includegraphics{path/to/image/myfig.pdf}   It would be:  \includegraphics{myfig.pdf}   While the image would still reside in the path/to/image/ directory. Basically, I'm asking if there is a macro that can recursively search for the image.  An alternative for me is to write a script outside of LaTeX that determines all the images that are included, performs the recursive search and copies those files temporarily in the same directory as the .tex file. How do I fix this "E: Type '*' is not known on line * in source list ..." update error? Every once in a while I see users having issues to update due to errors like this:  E:Type 'ain' is not known on line 1 in source list /etc/apt/sources.list.d/some-ppa.list'   with varying types/line numbers/source list files (often after removing a PPA).  How can such an error be fixed? Foreign key constraint on array member? Is it appropriate to use the salutation "Dear All" in a work email? I have observed that in my work place, whenever a mail is sent to more than one person( like an information, meeting request or a notice etc.), the mail starts with the salutation "Dear All". This, somehow, doesn't seem correct to me. For me, a salutation should address the reader individually, at least in its singular form, as in "Dear Colleagues".  Am I correct in thinking that? Headline Language Is there a particular term for the abbreviated language used in headlines (the removal of at least articles and conjunctions)? Use Stokes's Theorem to show $\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2$ I am a little stuck on the following problem:  Use Stokes's Theorem to show that  $$\oint_{C} y ~dx + z ~dy + x ~dz = \sqrt{3} \pi a^2,$$  where $C$ is the suitably oriented intersection of the surfaces $x^2 + y^2 + z^2 = a^2$ and $x + y + z = 0$.  OK, so Stokes's Theorem tells me that:  $$\oint_{C}\vec{F} \cdot d\vec{r} = \iint_{S}\operatorname{curl} \vec{F} \cdot \vec{N} ~dS$$  I have calculated:  $$\operatorname{curl} \vec{F} = -\vec{i} - \vec{j} - \vec{k}.$$  I then figured that on the surface $S$ we must have:  $$\vec{N}dS = \vec{i} + \vec{j} + \vec{k}dxdy$$  since this follows from the equation of the given plane.  However this will then give me:  $$\operatorname{curl} \vec{F} \cdot \vec{N} = -1 -1 -1 = -3$$  And thus I would get, if project this onto the $xy$-plane:  $$\iint_{S} \operatorname{curl} \vec{F} \cdot \vec{N} ~dS = -3 \iint_{A} dA = -3 \pi a^2$$  which is obviously not correct.  I would greatly appreciate it if someone could help me with this.  I actually had multivariable calculus a few years ago, and I know that I knew this stuff then.  However, now that I need it again I notice that I've become quite rusty.  Thanks in advance :) (Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems)  I'm aware of Sigma notation, but is there a function/name for e.g.  $$ 4 + 3 + 2 + 1 \longrightarrow 10 ,$$  similar to $$4! = 4 \cdot 3 \cdot 2 \cdot 1 ,$$ which uses multiplication?   Edit: I found what I was looking for, but is there a name for this type of summation? Spin or rotate an image on hover Do I have to leave the Schengen area if I have a short term (C) visa that will expire right before my National Visa (D) officially starts? If I have a Type C visa from the Norwegian embassy valid for February 9-28, and my Type D national visa from the German embassy valid from March 1 - August 31, 2017, do I need to leave/fly out of the Schengen area on the 28th and fly back on the 1st day of March? "No, I didn't see him" vs "Yes, I didn't see him" Bob didn't come to office today.  I was asked "Didn't you see Bob today?"  Should I say "Yes, I didn't see him today" or "No, I didn't see him today"? We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$  $a_1$ and $a_2$ are positive constants. We have to show the following:   The sequences $\{ b_n \} = \{ a_{2n-1} \}$ and $\{ c_n \} = \{ a_{2n} \} $ are monotonic, and if one is increasing, the other is decreasing. The limit of the sequence $\{ a_n \} $ is $ \left(a_1a_2^2\right)^{\frac13} $     Now, I have proved the first part. Besides that, I have also proved a few other things:  If $a_1 &gt; a_2$, then:  a. $ \{ b_n \} $ decreases, and $ \{ c_n \} $ increases.  b. $ c_n &lt; b_n $  If $a_1 &lt; a_2$, we just flip $\{ b_n \}$ and $\{c_n\}$  Besides, I have also shown that both the sequences : $\{b_n\}$ and $\{ c_n\}$ have the same limit. What I don't know, is how to evaluate the limit. Is it possible that galaxies' redshift is caused by something else than the expansion of space? I was thinking that maybe photons loss energy naturally when they travel great distances.  Or maybe the mass of all matter is increasing over time and therefore photons emitted in the past are necessarily less energetic.  Or is the expansion of space backed up by so much other evidences that trying to find another explanation is completely foolish? In the  question one of the answers had the following code snippet:  IEnumerable&lt;object&gt; FilteredList() {     foreach(object item in FullList)     {         if(IsItemInPartialList(item))             yield return item;     } }   What does the yield keyword do there?  I've seen it referenced in a couple places, and one other question, but I haven't quite figured out what it actually does.  I'm used to thinking of yield in the sense of one thread yielding to another, but that doesn't seem relevant here. How can I get very vivid colors in night cityscape pictures?