Definition: A real function $f:X\to\mathbb{R}$ is said to be bounded if there exists a $k&gt;0$ such that $|f(x)|\le k$      Definition: A subset $X$ of a metric space $(M, d)$ is said to be bounded if $d(x,y)\le c$ for all $x,y\in X$.   Let $p:\mathbb{R}\to \mathbb{R}$ a real polynomial function of degree $n&gt;0$ given by $$ p(t) = a_0 + a_1t+ \cdots + a_nt^n $$ Show that $p$ is not bounded. Moreover, show that for every bounded subset $X\subset\mathbb{R}$, the restriction $s:X\to \mathbb{R}, s(t) = p(t)$ is bounded.
Let $p(t) =  a_0+\dots +a_nt^n, a_n\neq 0,$ a polynomial with degree $n&gt;0$. Prove that $p:\mathbb{R}\rightarrow \mathbb{R}$ is not bounded, but for any bounded subset $X\subset \mathbb{R}$ the restriction $p\vert X$ is bounded.   I want to prove this in very elementary terms. I mean, without using the notion of limit or even continuity. I can only show this with "strong" results. Is it possible with elementary terms?
What is the www-data user? Related to Meta Super User efforts:  and Meta Server Fault efforts:  We already auto-remove negatively voted unanswered old questions automatically after 30 days, network wide, with no human intervention required. I was thinking we might extend that to remove old, unanswered zero score questions... based on the following criteria:  fewer than (question age in days * 1.5) views 0 score or lower, or 1 score or lower if the original author is deleted no answers 1 comment or fewer asked more than 1 year ago (thus based on creation date, not last activity date, so , or a user editing it, does not give a doomed question 365 more days of zombie &quot;life&quot;)  See the results of this query on:  (10509) |  (787) |  (650) This query has been refined based on the comments, and this is the final version that will roll out tonight. Every site in the network (except metas) will automatically delete old questions that meet this criteria from this point forward. Ambiguity of “not.... because” You are not hired because you are qualified.   I think this means you are hired but it is not because you are qualified. But I think this can  also means you are not hired, and the reason for it is that you are qualified.   Does this ambiguity really exist or do I mistake anything? Does a bad grade in one course (with otherwise OK record) affect graduate admissions? My /boot partition is nearly full and I get a warning every time I reboot my system. I already deleted old kernel packages (linux-headers...), actually I did that to install a newer kernel version that came with the automatic updates.  After installing that new version, the partition is nearly full again. So what else can I delete? Are there some other files associated to the old kernel images?  Here is a list of files that are on my /boot partition:  :~$ ls /boot/ abi-2.6.31-21-generic         lost+found abi-2.6.32-25-generic         memtest86+.bin abi-2.6.38-10-generic         memtest86+_multiboot.bin abi-2.6.38-11-generic         System.map-2.6.31-21-generic abi-2.6.38-12-generic         System.map-2.6.32-25-generic abi-2.6.38-8-generic          System.map-2.6.38-10-generic abi-3.0.0-12-generic          System.map-2.6.38-11-generic abi-3.0.0-13-generic          System.map-2.6.38-12-generic abi-3.0.0-14-generic          System.map-2.6.38-8-generic boot                          System.map-3.0.0-12-generic config-2.6.31-21-generic      System.map-3.0.0-13-generic config-2.6.32-25-generic      System.map-3.0.0-14-generic config-2.6.38-10-generic      vmcoreinfo-2.6.31-21-generic config-2.6.38-11-generic      vmcoreinfo-2.6.32-25-generic config-2.6.38-12-generic      vmcoreinfo-2.6.38-10-generic config-2.6.38-8-generic       vmcoreinfo-2.6.38-11-generic config-3.0.0-12-generic       vmcoreinfo-2.6.38-12-generic config-3.0.0-13-generic       vmcoreinfo-2.6.38-8-generic config-3.0.0-14-generic       vmcoreinfo-3.0.0-12-generic extlinux                      vmcoreinfo-3.0.0-13-generic grub                          vmcoreinfo-3.0.0-14-generic initrd.img-2.6.31-21-generic  vmlinuz-2.6.31-21-generic initrd.img-2.6.32-25-generic  vmlinuz-2.6.32-25-generic initrd.img-2.6.38-10-generic  vmlinuz-2.6.38-10-generic initrd.img-2.6.38-11-generic  vmlinuz-2.6.38-11-generic initrd.img-2.6.38-12-generic  vmlinuz-2.6.38-12-generic initrd.img-2.6.38-8-generic   vmlinuz-2.6.38-8-generic initrd.img-3.0.0-12-generic   vmlinuz-3.0.0-12-generic initrd.img-3.0.0-13-generic   vmlinuz-3.0.0-13-generic initrd.img-3.0.0-14-generic   vmlinuz-3.0.0-14-generic   Currently, I'm using the 3.0.0-14-generic kernel. Is the universe finite or infinite? Do the SO admins have any plans to release the underlying code for SO at all? I appreciate that the code may be part of their competitive advantage, but it strikes me that the underlying system would work for any topic, not just programming.  I am not advocating this, just curious really, but figured it might be under consideration as most of the value in SO is in the questions and answers, which have been released under a relatively open CC license. Book a return flight a year in advance Is it possible to buy an airline ticket that has me returning to my country after a year being abroad? I can't seem to book a flight that far in advance online. 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 Colors in Photoshop CS6/CC look different than in browser? One of my friends, a graphic designer told me recently that sometimes colors in Photshop look slightly different than the ones rendered in browser.  I know it is possible when you save a file and then use it on a website due to palettes, but he uses given color in PS, for example #b81e1e, then he creates a DOM element and sets its background to #b81e1e and the color is rendered differently in browser than in PS (I guess it's not that subtle as he can easily spot it). It's not a regular thing and it doesn't happen always, but he states it happened to him a few times, although he can't really tell me with witch colors.    Is it even possible? Because I have never really heard about anything like this &amp; nothing like this ever happened to me... Which versions of C++ redistributables can I remove? I have a number of Microsoft Visual C++ 2005 and 2008 Redistributable and I would like to know which ones are safe to remove, because there are actually more than 10 installed on my computers.   Microsoft Visual C++ 2005 ATL Update kb973923 - x64 8.0.50727.4053 Microsoft Visual C++ 2005 Redistributable Microsoft Visual C++ 2005 (x64) Microsoft Visual C++ 2008 ATL Update kb973924 - x64 9.0.30729.4148 Microsoft Visual C++ 2008 Redistributable - x64 9.0.30729.17 Microsoft Visual C++ 2008 Redistributable - x86 9.0.30729.17 Microsoft Visual C++ 2008 Redistributable - x86 9.0.30729.4148 Microsoft Visual C++ 2008 Redistributable - x86 9.0.30729.4974 Microsoft Visual C++ 2010 x64 Runtime - 10.0.30319   It could be that these versions are all different, but a confirmation of that is fine for me as well. Thanks in advance.  PS. I'm using an x64 machine one windows 7. How to evaluate the product: How to evaluate the product:  $$\prod _{n=2}^\infty\left(1+\dfrac{1}{n^2}+\dfrac{1}{n^4}+\dfrac{1}{n^6}+\dfrac{1}{n^8}+\ldots\right)$$  $$\left(1+\dfrac{1}{n^2}+\dfrac{1}{n^4}+\dfrac{1}{n^6}+\dfrac{1}{n^8}+\ldots\right)=\dfrac{1}{1-\dfrac{1}{n^2}}$$  How to proceed now? I have a proof and need some feedback. It seems really obvious that the statement is true but it is always the obvious ones that are a little trickier to prove. So I would appreciate any feedback. Thank you!  Here is what I am asked to prove:     If $n$ is composite then $(n-1)! \equiv 0 \pmod n$.   Proof:  $n$ is composite $\implies n=ab$ where $a,b \in \mathbb{Z}$ and $0&lt;a,b&lt;n$.  Case 1: If $a=b$ then $n=a^{2}$. Now $n \mid (n-1)! \implies a \mid (n-1)!$, so $$\begin{aligned} (n-1)! &amp;\equiv 1\times 2\times \dotsb \times a \times\dotsb\times (n-a)\times\dotsb\times (n-1) \\ &amp;\equiv 1\times 2\times \dotsb\times a \times\dotsb\times -a\times\dotsb\times -1 \\ &amp;\equiv 0 \pmod n \end{aligned}$$  Case 2: $0&lt;a&lt;b&lt;n$.  Then, since $a \mid n$, $b \mid n$ and $n \mid (n-1)!$ we have that $a \mid (n-1)!$ and $b \mid (n-1)!$.  So this implies $(n-1)! \equiv 1\times 2\times \dotsb\times a \times\dotsb\times b\times\dotsb\times (n-1) \equiv 0 \pmod n$, Q.E.D. Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$ Is there a nice presentation for the group $\text{PSL}(2,\mathbb{F}_q)$, for every prime $q$?  (I don't have any particular definition in mind for "nice", other than, say, a small number of generators and relations.) Connection between rank and matrix product I have a problem understanding the following:     Let $A$ be an $m \times n$ matrix and let t $\in \mathbb{N}$. Prove that   $\operatorname{rank}(A)\leq t$ if and only if there exists an $m \times t$ matrix $B$ and a $t \times n$ matrix $C$ so that $A = BC$.   I know what a rank is but I can't make a connection between the rank and the existence of two matrices such that $A = BC$. Is it possible to use JSF+Facelets with HTML 4/5? When electric dipole placed in non uniform electric field, what is the approach to calculate torque acting on it? Can it be zero? Is there a way to add an error on a dynamically referenced field? Let's take this simple example:  opp.get('Name').addError('Invalid name');   This is not working, I'm looking for a similar minded way.  Thanks Installing Reaver 1.4 I am trying to install reaver 1.4, but when I run ./configure ; make I get this message at the end  int ret_val = EXIT_FAILURE, r = 0;   I have searched a fair bit and cannot find an answer, any help would be appreciated. I apologize if this has been answered and I've just failed to find it. My actual WhatsApp back total size is 474Mb but when I go to I cloud back up it is showing 10GB, how is it possible? I mean I have deleted all media files from my WhatsApp and in WhatsApp application the total space is 474MB and also in iCloud storage it is 474MB, but when I entered further in to Backups and then choose my phone, it shows total backup size is 14GB and WhatsApp itself is taking 10GB. Can someone help with this?