Prove if a discrete subset $E$ of $\mathbb{R}^n$ is compact, then it is finite.
Discrete and compact subset must be finite
Behaviour of increment and decrement operators in Python Full-wave bridge rectifier with capacitor filter and ripple voltage According to various sources, the ripple voltage of a full-wave bridge rectifier is   $$ V_r = \frac{V_s}{2}fCR $$  where \$V_s\$ is the peak of the input voltage source, \$f\$ is the frequency, \$C\$ is the capacitor and \$R\$ the resistor.  I have simulated this circuit on PSPICE, circuitlab, and built it in real life:   \$R\$ is \$4.7K\$ and \$C\$ is \$10 uF\$. Voltage source is sine wave with \$0 V\$ offset, \$5V\$ amplitude and \$40 Hz\$ frequency.  All 3 experiments give me a ripple voltage of approximately \$0.65V\$, but the formula gives me \$1.330 V\$ (twice as much).  What am I missing? Evaluating $\int P(\sin x, \cos x) \text{d}x$ How often do ewoks eat sentients? In Return of the Jedi, the ewoks plan on eating Luke, Han and Chewie.     THREEPIO: I'm rather embarrassed, General Solo, but it appears you are to be the main course at a banquet in my honor.   At this point, the tribe had already met and befriended Leia, and had encountered the Imperials in the past, making it fairly likely that they knew humans were .  This raises the question of how often they ate sentients. Was it a one-off thing, to honor an extremely rare visit from a god, or was this fairly common? If light is an electric and (magnetic field), how can it be absorbed? I was wondering how light or any electromagnetic radiation can be "absorbed" if it consists of electric and magnetic fields. For example if there is a charge at point A, and the  light reaches point A, it will cause that charge to move depending on the value of the electric field. However, why wouldn't this electric field continue on and influence others particles everywhere else? For example, if you have a static charge creating an electric field, then if you put a charge at point A it will feel its effects and then if you bring a charge at another location B it will also feel its effects, namely it will not have been absorbed. Please shed some light on this. Thanks  edit: Does this mean the electron that absorbs this light moves in such a way that it creates an electric field of its own canceling the initial one? Or am I way off? Is it possible to check for an injured player in Minecraft? I am creating a mini-game in Minecraft, where I must check to see if the player is injured. The injured player will then be given an item. My guess is that the command would look something like /testfor @p[injured=true], but this command doesn't work. Is there a command for this? Proof: in $\mathbb{R}$, $((0,1),|\cdot|)$ is not compact. Let $(M,d)$ be a metric space, and $A\subset M$. By definition, $A$ is said to be compact if every open cover of $A$ contains a finite subcover.  What is wrong with saying that, in $\mathbb{R}$, if $I=(0,1)$, we can choose $G=\{(0,\frac{3}{4}), (\frac{1}{4}, 1)\}$, which satisfies $I \subset \bigcup_{U\in G} U$, but we can't extract a finite subcover, so $I$ is not compact. Is $G$ a finite subcover of $G$, so it is not a valid cover for proving this? I would take $\cup_{n\in\mathbb{N}} (\frac{1}{n},1)$ in order to prove this, can we conclude that every open cover is necessarily a infinite union of open sets $\neq \emptyset$? Reference book on measure theory I post this question with some personal specifications. I hope it does not overlap with old posted questions.  Recently I strongly feel that I have to review the knowledge of measure theory for the sake of starting my thesis.  I am not totally new with measure theory, since I have taken and past one course at the graduate level. Unfortunately, because the lecturer was not so good at teaching, I followed the course by self-study. Now I feel that all the knowledge has gone after the exam and still don’t have a clear overview on the structure of measure theory. And here come my specified requirements for a reference book.   I wish the book elaborates the proofs, since I will read it on my own again, sadly. And this is the most important criterion for the book. I wish the book covers most of the topics in measure theory. Although the topic of my thesis is on stochastic integration, I do want to review measure theory at a more general level, which means it could emphasize on both aspects of analysis and probability. If such a condition cannot be achieved, I'd like to more focus on probability. I wish the book could deal with convergences and uniform integrability carefully, as Chung’s probability book.    My expectation is after thorough reading, I could have strong background to start a thesis on stochastic integration at an analytic level.  Sorry for such a tedious question.  P.S: the textbook I used is Schilling’s book: measures, integrals and martingales. It is a pretty good textbook, but misprints really ruin the fun of reading. I'd like to find an analogue of  or any other fps counter on Ubuntu 12.10.  After some googling I found 2 solutions:   To use  (I found it from ). This solution didn't work for me because after launching the game:  mumble-overlay game   I got this error:  game: symbol lookup error: /usr/lib/mumble/libmumble.so.1: undefined symbol: glPopClientAttrib   And I didn't find any fix of this error in the WEB. To use WINEDEBUG=fps with osd_cat. It works only with  (only Windows games). Found from reading  and . It shows FPS during play but the output is definitely wrong - 0.15-0.18 fps. And it doesn't matter if game really runs slow or fast - the result is always the same.   Does anyone know how to fix these errors? Or are there any other solutions? Thanks for the help. Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$ Let $X=C[0,1]$ be the set of all continuous functions on $[0,1]$. For any two functions $f,g\in X$, set $$d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|.$$  Prove $(X,d)$ is a metric space.    My attempt:  This question comes down to showing that $d$ is a metric, which means that:   $d(f,g)\geq 0$ $d(f,g)=0\iff f=g$ $d(f,g)=d(g,f)$ Triangular sadness.     1) Nonnegativity is guaranteed by taking the supremum of the absolute value of a sum, since the absolute value of a sum is non-negative, the supremum of non-negative numbers is non-negative.  $\square$    2) For the supremum of $|f(x)-g(x)|$ to be equal to $0$, the greatest number for $|f(x)-g(x)|$ from $0\leq x\leq 1$ must be $0$, hence $|f(x)-g(x)|=0$ for all values on this range, and thus $f(x)=g(x)$ is the constant zero map.  $\square$    3) $|f(x)-g(x)|=|g(x)-f(x)|$ implies that $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|=\sup \limits_{0\leq x \leq 1} |g(x)-f(x)|$ $\square$    4) Triangular sadness:  We need to prove that $d(f,g)\leq d(f,h) + d(g,h)$  I.e.  $\sup \limits_{0\leq x \leq 1} |f(x)-g(x)|\leq \sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$  Now I am not sure how I can 'behave' with the sup 'function'. Can I do the following?  $$\sup|f(x)-g(x)| = \sup|f(x)-g(x)+h(x)-h(x)|=\sup|f(x)-h(x)+(h(x)-g(x))|$$ Then use normal triangular inequality:  $$\sup|f(x)-h(x)+(h(x)-g(x))|\leq \sup(|f(x)-h(x)|+|h(x)-g(x)|)=\sup(|f(x)-h(x)|+|g(x)-h(x)|)$$   This is obviously less than $\sup \limits_{0\leq x \leq 1} |f(x)-h(x)|+\sup \limits_{0\leq x \leq 1} |g(x)-h(x)|$, but I don't know how to prove it. How do I finish that one off. Let me know I have upvotes even after hitting the rep cap When you cap, you don't see any upvotes you get on your profile, therefore, you don't have any update on how your questions / answers are doing unless you look at them manually.   Capping the 200 daily rep is not a reason to stop posting here so could I get notified somehow that my posts are being voted on ? Maybe put a little +X in a different color letting me know that I got these votes but I won't get the reputation for it.   I can go and see in my reputation feed inside my profile and see it :    So I don't see why it wouldn't show up in the feed above.  Maybe putting the feed button green without any +X just like when you aquire a badge. Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$ I tried many approaches but none of them really worked I treated $p^b-1$ as a Geometric progression but it didn't work and that is about as far as I have been able to go I have no clue how to move forward Creating a link using an 18 character ID I have an 18 character SFDC ID. I was under the impression that I should be able to create a link to that record using the following pattern: https://&lt;org server&gt;.salesforce.com/&lt;id&gt;  For example: https://na12.salesforce.com/00TU000000Nf5NP  That link works fine for me. However, when I try to link to the same object using the 18 character, case insensitive ID (ex. https://na12.salesforce.com/00tu000000nf5npmaz), I get the following error:     Insufficient Privileges      You do not have the level of access necessary to perform the operation   you requested. Please contact the owner of the record or your   administrator if access is necessary.   What exactly is the problem? If all I have is the 18 character ID, how can I create a link back to the correct SalesForce record? I just got a new hat! :D screaming with joy Looking at so many users looking all exquisite with their stunningly 'fitted' hats, I'm wondering if there's an option to resize or rotate a hat?  PS: Happy Winter Bash everyone! :) I recall having heard somewhere that the only 1-manifolds (second countable, Hausdorff, connected spaces locally homeomorphic to $\mathbb R$) are $\mathbb R$ and $S^1$. Is this true? If so, is there a reasonably elementary proof? Short story in book collection about war machines and armies out of control I read in an anthology book in 1975. It starts with two people watching a “robot” plane glide high and then dive into an opening of a factory causing a massive explosion. I’m not sure but I think the people may have orchestrated the attack on purpose to get the robot armies fighting each other in the hope that it would diminish robot army capabilities and bring them under control. The robot wars did occur, but at the end of the story the people discover a tube sticking out of the ground that is periodically shooting something into the air. Nearby they find a canister that did not get launched out of the “cannon” successfully and it was broken open. After close examination they discover tiny “ants” on the canister that are actually tiny robots which are rebuilding the robot war factories. The robots have adapted and will keep trying to fulfill their mission. This story may  have been in a booked edited by Roger Elwood, but I cannot guarantee it. strange character replacement on SO I added a comment to  but something strange is happening. The end code reads "); but SO decided to change that to ";);.    I did not add a ; in front of the ).  To further explain what's happening here, the following:  "http://example.com/." http://example.com/. "http://example.com." http://example.com. "http://example.com" http://example.com   renders as: What is the best way to save values (like strings) for later use? I'm fairly new to LaTeX and I'm struggling with something apparently simple, but for which I can't find a fair solution. The problem (simplified) is as follows (please take care terms used may not have the traditional meaning in LaTeX):  I need to references to files (many) throughout a text. As example, the files:  "C:/Work/this_file1.c", "C:/MoreWork/this_file1.c" and "D:/MoreWork/another_file2.h".  So, now I would like to create an "alias" to each directory and each name file so I can reference them. A "pseudocode" could be:  % Definition \alias \c_dir = C:/Work/ \alias \d_dir = D:/MoreWork/ \alias \file1 = this_file1.c \alias \file2 = another_file2.h  % Reference in text \c_dir\file1    % For referencing C:/Work/this_file1.c \d_dir\file1    % For referencing D:/MoreWork/this_file1.c \c_dir\file2    % For referencing C:/Work/another_file2.h   My first approach was using \newcommand, but that caused numerous problems due to naming limitations. (I also tried other methods like pgfkeys, but it always gets stuck somewhere!)  Can you suggest a method? It is stated as a problem in Spivak's Calculus and I can't wrap my head around it. Questions about right ideals