Let f be a function from real numbers to real numbers and let f(x) be a continous function, such that:  $$\lim_{x\rightarrow\infty}f(x)$$ $$\lim_{x\rightarrow-\infty}f(x)$$  are bounded.  How do I prove that f(x) is bounded?
$f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function. Assume that $\lim_{x \rightarrow \pm \infty} f(x)$ exist and are finite. Prove that $f$ is bounded.  So to show that $f$ is bounded, I must show that $\exists M \in \mathbb{R}$ such that $|f(x)| \leq M$ for all $x \in Dom(f)$. Assuming that $f(x)$'s limit exist, then I know that $\forall \epsilon  &gt; 0$, $\exists N$ such that for $n &gt; N$, $|f(n) - L| &lt; \epsilon$. But I am having trouble with figuring out how to connect the two ideas together.
Installing .pkg with terminal ? Is it possible to install a .pkg using a terminal ? (I wanted to install an app through ssh). What is the plural of the abbreviation of "multiplicity automaton", "MA" or "MAs"? The "multiplicity automaton (MA)" is a model in compute science and its plural is "multiplicity automata". Should the plural of the abbreviation be MA or MAs?      is not a duplicate as it is focused on cases like "ATMs" where the expanded form "automated teller machines" has a regular plural ending in "s." Unlike "machines," the word "automata" has an irregular plural that does not end in "s." It's unclear from the linked answers how abbreviations that end in words like this should be pluralized. A high-ranking  mentions "VIPs," but a high-ranking  says     VIPs' is a strange one. It could be argued that the expanded acronym,   when pluralized, should be 'Very Important People' rather than 'Very   Important Persons'. Using this argument, 'VIP' could be used as a   singular or plural noun. "The VIP has arrived." "The VIP have   arrived." I'm not sure anyone else would agree with my logic, though. If you run a code snippet, a hat still appears (hats are above snippets in the Z-order) Does the Dueling Fighting Style prevent using a shield? The Dueling fighting style doesn't seem to state anything about shields, stating that the character can only wield one weapon while fighting to gain the benefit. Does a shield count as a weapon or otherwise stop the damage bonus from occurring? Photoshop layers into SVG paths If $\lambda$ is an eigenvalue of $A^2$, then either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A$ $A$ is an $n\times n$ matrix of complex numbers. Prove that if $\lambda$ is an eigenvalue of $A^2,$ then $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A.$    If $\lambda$ is an eigenvalue of $A^2,$ we have $\lambda\alpha=A^2\alpha$ for some $\alpha.$ Then how can we find a $\beta$ s.t. $\sqrt{\lambda}\beta=A\beta?$ I have a site on my webhotel I would like to run some scheduled tasks on. What methods of achieving this would you recommend?  What I’ve thought out so far is having a script included in the top of every page and then let this script check whether it’s time to run this job or not.  This is just a quick example of what I was thinking about:  if ($alreadyDone == 0 &amp;&amp; time() &gt; $timeToRunMaintainance) {    runTask();    $timeToRunMaintainance = time() + $interval; }    Anything else I should take into consideration or is there a better method than this? Measures of variable importance in random forests I've been playing around with random forests for regression and am having difficulty working out exactly what the two measures of importance mean, and how they should be interpreted.  The importance() function gives two values for each variable: %IncMSE and IncNodePurity. Is there simple interpretations for these 2 values?  For IncNodePurity in particular, is this simply the amount the RSS increase following the removal of that variable? How do you get Mac OS Finder to group folders together when sorting? I have Mac OS 10.7.4 and I'd like Finder to use "Is Folder?" as its first sort criteria, or in other words, group folders before files regardless of the sorting method.  Windows and Linux both do this by default. Single word describing a person new in something What is a word for someone who is totally new in something or participates in something for the first time? So, somehow over the weekend, my 1GB USB flash drive has managed to become write-protected.  There is no switch on the stick, so I'm concluding that something has corrupted at some point.  Obviously I can't format it or remove.  The output from dmesg | tail is:  Buffer I/O error on device sdb1, logical block 328168 lost page write due to I/O error on sdb1 sd 73:0:0:0: [sdb] Unhandled sense code sd 73:0:0:0: [sdb] Result: hostbyte=DID_OK driverbyte=DRIVER_SENSE sd 73:0:0:0: [sdb] Sense Key : Data Protect [current] Info fld=0x0 sd 73:0:0:0: [sdb] Add. Sense: Write protected sd 73:0:0:0: [sdb] CDB: Write(10): 2a 00 00 05 08 00 00 00 01 00 Buffer I/O error on device sdb1, logical block 328200 lost page write due to I/O error on sdb1   I think the second and final lines are a giveaway to corruption, but I don't know how to interpret this output.  Could someone help to see if this output gives any indication?  I usually eject the drive before removing it, but I may have been lax once or twice.  The data isn't critical, it's just bugging me.  Any suggestions on how to fix it (if the drive even can be) would be greatly appreciated. Generate random numbers uniformly over an entire range Cannot set up multiple desktops (workspaces) "Unfortunately, ... has stopped" when Downloading file in Android How to delete a file ending in a dot in Windows 7? After installing Cygwin on Windows 7 and realizing I'm better of with using Linux on a different partition, I wanted to uninstall it. However, Cygwin has no uninstaller so I was left with the option of deleting the whole folder it was installed in – which I did.  The problem is that a single file remained, in the whole directory tree, named README. at  C:\cygwin\usr\share\texmf-dist\doc\latex\hausarbeit-jura\README.  I know Windows isn't supposed to allow files with names ending in ., but this one obviously got through, and I cannot delete it. When trying to do that, I get     Item Not Found      This is no longer located in C:\cygwin\usr\share\texmf-dist\doc\latex\hausarbeit-jura. Verify the item's location and try again.   When viewing the properties of the file (Right-click, Properties), Windows tells me that it has 0 bytes.   However, when viewing it in the explorer window, and clicking on it, the windows status bar thing tells me it has 517 bytes.  Any ideas on how to get rid of it? I've tried searching for this question but couldn't find it on stackexchange. This is a common type of interview question; I ran into it doing brain teasers on a probability puzzles app, and if you fine people agree with my logic, I will inform the app developer that his/her answers are incorrect.   The problem is essentially this:  You are sentenced to death for thievery. The King is magnanimous and decides to put your fate in the hands of chance. You are given $100$ white marbles and 100 black marbles, and $2$ urns. The king will choose an urn at random and pull out a single marble at random; if the marble is white, you live, if its black, you die. If you place the marbles in the best way possible, what is your probability of survival?  I started with the base case: $100$ white marbles in one urn, $100$ black marbles in the other. This comes down to a $50$-$50$ chance of survival. I then worked my way to deciding that placing $1$ white marble in one urn and $99$ white marbles + $100$ black marbles in the other urn would be the "best way possible", which yields the following:  $$P(\text{Survival}) = \frac{1}{2}(1+\frac{99}{199}) \approx .749$$  Selecting $1$ of $2$ urns at random gives $\frac{1}{2}$, the urn containing $1$ marble gives $1$, and the other that contains $99$ white marbles and $100$ black marbles gives $\frac{99}{199}$ because there are $99$ possible white marbles to select out of $199$ total marbles.  The app claims that the correct answer is $\frac{1}{2}(1+\frac{99}{200}) \approx .748$   I see where the $200$ comes from, but I do not think it is right to say that there are $200$ marbles in the other urn. Who is correct? Could you tell me how to prove that if two series $ \sum_{n=0} ^{\infty}x_n, \sum_{n=0} ^{\infty} y_n$ are convergent, then $\sum_{n=0} ^{\infty}(\alpha \cdot x_n + \beta \cdot y_n)$ is also convergent and its sum equals $\alpha \cdot \sum_{n=0} ^{\infty} x_n + \beta \cdot \sum_{n=0} ^{\infty} y_n$? I normally use the following idiom to check if a String can be converted to an integer.  public boolean isInteger( String input ) {     try {         Integer.parseInt( input );         return true;     }     catch( Exception e ) {         return false;     } }   Is it just me, or does this seem a bit hackish?  What's a better way?    See my answer (with benchmarks, based on the  by ) to see why I've reversed my position and accepted  to this problem.  I think this original code will be used by most people because it's quicker to implement, and more maintainable, but it's orders of magnitude slower when non-integer data is provided. I started learning Bash a couple of days ago.  I'm trying to obtain an exit status of grep expression into a variable like this:  check=grep -ci 'text' file.sh   and the output that I got is   No command '-ic' found   Should I do it with a pipe command? Hiding reputation and encouraging thinking time would improve the quality of answers on SO I've only been a member of SO for a couple of months but I must say that the site can be a bit daunting even for an experienced developer.  Part of this is because when users with very high reputations answer questions then it seems that everyone assumes that they have the right answer.  I don't want to go into the self-fulfilling nature of reputation increase for high reputation people.  But I think it is clear that people with high reputation scores get more upvotes () because people assume (usually accurately) that they are correct.    But if reputation was hidden on SO, at least for, say, the first hour after a question is asked then the probability of the best answer being upvoted the most would surely increase.  This might also require the hiding of the names of the answerers if they have very high reps!  As a second method of getting improved answers I would correlate the value of an answer (in terms of reputation gained) with the difficulty of the question.  And the simplest way to do that would be based on the time required for the question to be answered.  So if a question is answered within a minute then it is a 20% question (2 points per upvote), within 2 minutes a 40% question, within 5 minutes a 60% question and so on, on to say a 200% answer (20 points per upvote) on questions that have been hanging around for months.  This would encourage considered answers to new questions and tempt people into reducing the backlog - rather than encouraging people to sit on the 'newest questions' hoping for a gimme.  These two measures would incentivise people to answer the questions that no-one likes the look of and reduce the relative value of answering easy questions.