Let $f\in C(\mathbb{R})$  be a function that satisfies $\displaystyle\lim_{x\to\pm\infty}f(x)=+\infty$. Show that $f$ attains its absolute minimum, i.e. there is a point $x_0\in\mathbb{R}$ such that $f(x)\geq f(x_0)$ for all $x\in\mathbb{R}$.  I was thinking about studying the set $M=\{L\in\mathbb{R}: f(x)\geq L, \forall x\in\mathbb{R}\}$. But I'm not sure if this is the right way to proceed.
If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $$\lim_{x\to\infty}f(x)=\lim_{x\to-\infty}f(x)=\infty$$  Prove that $f$ has a minimum on $\mathbb{R}$; that is, there is an $x_m\in\mathbb{R}$ such that $$f(x_m)=\inf_{x\in\mathbb{R}}f(x)&lt;\infty$$  I think the best approach would be to prove the existence of a lower bound, that is $\exists M\in\mathbb{R}:M&lt;f(x),\forall x\in\mathbb{R}$, but I don't know how to do this. I am probably supposed to use the extreme value theorem or the intermediate value theorem, but I don't know how to do that.
Existence of a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot)$ Let $(\mathbb K,+,\cdot)$ be a field. Is there a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot) $ ?  The answer should clearly be negative. I tried to proceed via contradiction, but this has not led me very far. Since it's some kind of &quot;trick problem&quot;, I'm merely looking for hints. Cross Browser Flash Detection in Javascript Does anyone have an example of script that can work reliably well across IE/Firefox to detect if the browser is capable of displaying embedded flash content. I say reliably because I know its not possible 100% of the time. Let us say we have the following Makefile  include config.mk  # Generate summary table. results.txt : $(ZIPF_SRC) isles.dat abyss.dat last.dat     $(ZIPF_EXE) *.dat &gt; $@  # Count words. .PHONY : dats dats : isles.dat abyss.dat last.dat  %.dat : books/%.txt $(COUNT_SRC)     $(COUNT_EXE) $&lt; $*.dat  .PHONY : clean clean :     rm -f *.dat     rm -f results.txt   where all the variables are defined in config.mk  LANGUAGE=python COUNT_SRC=countwords.py COUNT_EXE=$(LANGUAGE) $(COUNT_SRC) ZIPF_SRC=testzipf.py ZIPF_EXE=$(LANGUAGE) $(ZIPF_SRC)   Is there any tool available that can expand all the include lines within the makefile so as to produce a bigMakefile? If such a tool does not currently exist, can a python/perl/awk/bash script be cooked up to achieve this goal? What Happens to electrons after Alpha Decay and Nuclear Fission? Where do the electrons go? In alpha decay do 2 electrons follow the alpha particle and make stable Helium or does the larger daughter nucleus become an anion?  Also what do the electrons do in the mixture of fission and alpha decay? With Beryllium-8 it decays into two alpha particles and 4 lonely electrons or do you get two Helium atoms?If someone could add an entry to Holocron it would be greatly appreciated, thanks. Uses of the 'Golden Ratio' I have heard much about the numerous appearances of the ratio found in nature: 1.6180339887.  Are there any actual mathematical uses that have been found of this number? What are its advantages? Just curiosity, really. Non-standard signature of main() compiles successfully $ d_1,d_2\mid n\iff {\rm lcm}(d_1,d_2)\mid n\ $ [LCM Universal Property] Is there any way to create a bootable Ubuntu USB flash drive from the terminal without using any third-party applications like , , ,  etc.  I tried to create a bootable Ubuntu flash drive with dd method,  sudo umount /dev/sdb sudo dd if=/path/to/ubuntu.iso of=/dev/sdb bs=1M   It creates files on the USB disk, but when I try to boot the USB disk it shows an Operating System Not Found error. Alignment issue in 'Your Sites' section in the Android app When I cut a layer using marquee tool, I get blur edges to cut side. How do I get sharp edges after cut? I am using rectangular marquee tool to cut a portion of the layer in photoshop cs5, but after cutting, the edges are not sharp, I get blur edges.      how can I get sharp edges? I am in Portugal (Lisbon) at the moment, and when leaving my room late at night (12 pm ~ 04 am) I usually face about 4 of these arthropods in the corridors:    (For reference, the "line" in the image is a standard separation on the floor. This particular example of the animal is a little smaller than a nail.) They seem to have quite variable sizes and walk primarily on ground. Also, as implicit above, they seem to be active primarily during the late night. (I don't see them much often, if at all, during the morning/afternoon.) They can be quite fast, but if left unharmed do not move much. They also appear to take shelter inside the walls.  From what I searched, it seems possible that they are centipedes. I've seen sites saying that they are useful to keep other less pleasant animals at bay. My questions are: What are those animals, and should I leave them be or not? Line numbering paragraph oriented? I'm using the lineno package with the pagewise option but I need a way to get the numbering restart every paragraph break. I need a general solution to pass to the latex command, e.g.:  latex "\AtBeginDocument{\RequirePackage[pagewise]{lineno}\linenumbers}\input{mydocument.tex}"   I read the lineno manual but the package does not provide this feature. Is there any trick to do that? If I fall off a cliff - say, 1000 feet - can I wait (readied action?) until I am near enough to the ground (my understanding is you fall 500 ft/round) then  safely to the ground? Or, does the momentum from the fall carry over through the teleport and I still splatter across the rocks? AlwaysOn AG: Current Host Server - Owner Node - Primary Instance Solve simple trig equation When should com­pound words be writ­ten as one word, with hy­phens, or with spaces? Some compound words are written without hyphens (nonaggression, nonbeliever), some with hyphens (well-intentioned), and others with spaces (post office).  Is there a rule or good guide as to which option should be used? Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, including $L$ and $\mathbb{Q}$ as intermediate fields. I'm told that the fundamental theorem of galois theory can be useful here, but I can't find a use for it. Given a list ["foo", "bar", "baz"] and an item in the list "bar", how do I get its index (1) in Python? Sort and count number of occurrence of lines I have Apache logfile, access.log, how to count number of line occurrence in that file? for example the result of cut -f 7 -d ' ' | cut -d '?' -f 1 | tr '[:upper:]' '[:lower:]' is  a.php b.php a.php c.php d.php b.php a.php   the result that I want is:  3 a.php 2 b.php 1 d.php # order doesn't matter 1 c.php For equal cost, why would you select the MacBook Air over the MacBook Pro? I can see that the 13" version of MacBook Pro and MacBook Air is same $1199, but the Pro is more feature-loaded than Macbook Air. Why would anyone buy MacBook Air? What's the catch in this?    I was thinking of buying a MacBook for development purposes, and worried if there's any problem in the Pro version, because it comes with more features (like more storage).  Edit:    Thanks for the answer guys, I bought a MacBook Pro a week ago and already loving it.