I need to prove one thing: $$ n \ge 1: \sqrt{n} \le \sqrt[n]{n!} \le \frac{n + 1}{2} $$  The second part: $$ \sqrt[n]{n!} \le \frac{n + 1}{2} $$ is easy to proof.But the first is more complicated. Help please.
Show that if $n&gt;2$, then $(n!)^2&gt;n^n$.     My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n&gt;(n+1)^{n-1}$ Here, I tried to use induction again without any luck. I also took log of both sides, but I did not get anything. Please help!
How do I post code in chat? I would like to be able to post code in chat. I know that I can post inline snippets of code by using backticks, but I don't know how to post blocks of code, with the entire chat message formatted as code, and indentation respected. I am aware of . After reading the comments, I found out that I could format in "fixed width", but I can't see a "fixed width" button anywhere. What should I do? Redshifting of Light and the expansion of the universe So I have learned in class that light can get red-shifted as it travels through space. As I understand it, space itself expands and stretches out the wavelength of the light. This results in the light having a lower frequency which equates to lowering its energy.  My question is, where does the energy of the light go? Energy must go somewhere!   Does the energy the light had before go into the mechanism that's expanding the space? I'm imagining that light is being stretched out when its being red-shifted. So would this mean that the energy is still there and that it is just spread out over more space? I just checked my server's /var/log/auth.log and found that I'm getting over 500 failed password/break-in attempt notifications per day! My site is small, and its URL is obscure. Is this normal? Should I be taking any measures? dollar-terminated strings In my assembly language class, our first assignment was to write a program to print out a simple dollar-terminated string in DOS.  It looked something like this:  BITS 32     global _main  section .data     msg db "Hello, world!", 13, 10, ’$’  section .text _main: mov ah, 9 mov edx, msg int 21h ret   As I understand it, the $ sign serves to terminate the sting like null does in C.  But what do I do if I want to put a dollar sign in the string (like I want to print out "it costs $30")?  This seems like a simple question, but my professor didn't know the answer and I don't seem to find it using a google search. I am planning an hour and a half ride with my 1 month old baby girl. I heard that being in the car seat for a long time is not good for newborns.  Is an hour and a half too long? Should we do a stop in the middle of the trip? I'm trying to use biblatex and when I compile my document I get an error like the following:  Windows  data source C:\Users\&lt;username&gt;\AppData\Local\Temp\par-5061756c\ cache-890efc00b3ca6b775c7d44a325c1349fb2a3a3bd\inc\lib/Biber/LaTeX/recode_data.xml not found in .   Mac  data source /var/folders/v2/rld0ls7d2935gkvqv5hfr1x00000gn/T/par-616c616e/ cache-cdd483146f82a9655ce063f848d5139480fbf872/inc/lib/Biber/LaTeX/recode_data.xml not found in .   Linux  data source /tmp/par-6963617269756d/cache-f37ab610b7d79b2720a8ee3732849c6821705520 /inc/lib/Biber/LaTeX/recode_data.xml not found in . Command-line completion from command history So, I've looked at history and at Ctrl+R, but they are not what I thought I knew. Is there a way that I can type in the beginning of a command, and cycle through the matches in my history with some bash shortcut?  # mysq(some shortcut key)   Gives me:  # mysqldump  --add-drop-table -e -q -n -C -u  (some shortcut key) #  mysql -u ben.dauphinee -p What does the `who` command result `root :0` mean? The return value of command who will always include root :0 in X Window pts. I guess it actually means the GNOME login. Am I right?  Another question is where to get this knowledge, as I don't see it in the man who, neither in info who.  Example  [root@localhost root]# who root     :0           Sep 11 06:33  root     pts/0        Sep 11 06:36     (:0.0) How can we get "Developer" bugs logged with Technical Support? I recently found a bug, which I mentioned on , as well as submitting a case, and after a few days fumbling around with the case, they closed it by saying that they only provide this type of support for Premier Support customers.  Is there a better channel for reporting bugs than by submitting a case, or is there something "special" we have to do? I realize this question is similar to , except that I've already gone the route of , as suggested by the answer there (i.e. submitting a case), to no avail.  How can I make sure that a bug like this can find its way to someone capable of at least confirming that the behavior was expected? This behavior isn't documented anywhere, and so I assume that it's a bug that needs to be addressed. prove that if $E$ is connected and $E \subseteq F \subseteq \overline{E}$, then $F$ is connected. Define a set $A$ to be disconnected iff there exist nonempty relatively open sets $U$ and $W$ in $A$ with $U\cap W = \emptyset$ and $A = U\cup W.$ Define a set $A$ to be connected iff it is not disconnected.(there are many equivalent definitions, but I want to prove this lemma using this one). Prove that if $E$ is connected and $E\subseteq F \subseteq \overline{E},$ then $F$ is connected.  Let $U, W$ be a separation for $F$. Find open sets $O_U$ and $O_W$ so that $U = F \cap O_U$ and $W = F\cap O_W.$ I claim that $E\cap O_U, E\cap O_W$ separate $E$. However, I'm unable to show that $U' = E\cap O_U, W' =E\cap O_W \neq \emptyset$ (I think this should be straightforward, but for some reason, I can't figure this out). Suppose $U' = \emptyset.$ Then $E\cap O_U = \emptyset.$ Since $E \subseteq F = U\cup W = F\cap (O_U \cup O_W)\subseteq O_U\cup O_W,$ we have that $E \subseteq O_W,$ so $E\cap O_W = E\subseteq F\cap O_W = W\subseteq F\subseteq \overline{E}.$ Observe that since $E\cap O_U = \emptyset, F\cap O_U = (F\backslash E)\cap O_U\subseteq F\backslash E.$ Similarly, $W'\neq \emptyset.$ Clearly, $U', W'$ are relatively open in $E$. Suppose $U'\cap W' \neq \emptyset.$ Let $x\in U'\cap W'.$ Then $x\in E\cap O_U\cap O_W \subseteq F\cap O_U\cap O_W = U\cap W = \emptyset,$ a contradiction. So $U'\cap W' = \emptyset.$ Also, $U'\cup W' = (E\cap O_U)\cup (E\cap O_W) = E\cap (O_U \cup O_W)$ and $E\subseteq (O_U \cup O_W),$ so $U'\cup W' = E.$ Why is there an emission of gamma rays? When a spontaneous radioactive reaction happens, there is an emission of gamma rays (in most cases) What causes this emission? How to disconnect selected mesh geometry? I don't know how to disconnect a face from the rest of the mesh in edit mode.  This isn't like pressing P to separate the selection into a completely different object. I just want to select a face and disconnect/rip it away from the rest of the mesh but keep it as a part of the original object.  One way to do it would be to duplicate the face with ShiftD and then delete the original face. But I suspect there is a "native" command for disconnecting selected faces and I would like to learn it. How can I install MonoDevelop v4 on 12.04? I am using Ubuntu 12.04. I can install v2.8.6.3 of Monodevelop from software center but I need at least v3 to use Monogame v3 or higher. Can anyone point me in the right direction please?   Thanks In a conference on elliptic curves (an introduction to the subject), the speaker said that an elliptic curve (I.e. an equation of the form $y^2=x^3+ax+b $ where the RHS has distinct roots) is, in the complex space, a torus/Riemann surface of genus 1.  What is meant by that? Are we talking about a 2-dimensional manifold the 4D space? Export EDITOR with a flag I'm using Zsh and some CLI's such as Google's gcloud load files via $EDITOR.    For me, I have set this as:  export EDITOR="/usr/local/bin/mvim"   With aliases for that editor using the -v flag which loads vim in the terminal:   mac_vim="$EDITOR -v"  alias vi=$mac_vim  alias vim=$mac_vim  alias v=$mac_vim   That works but when the gcloud CLI calls $EDITOR it does not have the -v flag so it opens in a new terminal.  I thought I could change that by instead adding the -v flag here:  export EDITOR="/usr/local/bin/mvim -v"   However, when I try and run that on the command line I get:  $EDITOR text.html zsh: no such file or directory: /usr/local/bin/mvim -v   Is it possible to include a flag when exporting my $EDITOR? 70s or 80s Book. Post Catastrophe, Sheriff tries to hold town together, and keep his schizo wife from killing him I have a directory that I am trying to clean out that contains both files and subdirectories.  What I am trying to do is simple: move all the files into another directory, but leave all the sub-directories as they are.  I am thinking something like:  mv [*_but_no_dirs] ./other_directory   It seems like there should be a simple way to do this with wildcards * and regex...  Anyone have ideas? Multi armed bandit for general reward distribution I'm working on a multi-armed bandit problem where we do not have any information about the reward distribution.   I have found many papers which guarantee regret bounds for a distribution with known bound, and for general distributions with support in [0,1].   I would like to find out if there is a way to perform well in an environment where the reward distribution does not have any guarantees about its support.  I'm trying to compute a nonparametric tolerance limit and using that number to scale the reward distribution so I can use the algorithm 2 specified on this paper (). Does anyone think this approach will work?  If not, can anyone point me to the right spot?  Thanks a bunch! Difference between `dpkg -i` and `apt-get install` From all the other Q&amp;A (like: ) I understand that apt-get is a wrapper over dpkg and provides dependency management.  Assuming I'm installing something that doesn't have any dependencies (or they're also installed at the same time) would there be any difference between installing it using apt-get vs dpkg  E.g. (kinda concrete differences I'm looking for)   say dpkg can not download the pkg from somewhere, I must download before starting installation, whereas apt-get will do this on my behalf. dpkg won't configure pkg, it'll only copy files. apt-get will configure too.   ... Why $\gcd(qb+r,b)=\gcd(b,r)$?