Help I'm really stuck on a proof question:  Prove that, for all positive integers for $x$ and $y$, $\sqrt{ xy} \leq \frac{x + y}{2}$   Thanks!
I am having trouble with this problem from my latest homework.  Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x,y$, we have $$ \sqrt{xy}≤ \frac{x+y}{2} .$$ Furthermore, equality occurs if and only if $x = y$.  Any and all help would be appreciated.
What is IEEE 802.3 Ethernet's communication type: Synchronous or Asynchronous? In the IEEE 802.3 frame structure there is a 7 byte preamble section for synchronization. If it is really for synchronization does that mean Ethernet is synchronous and not asynchronous?    Ethernet structure for reference: I have put a bounty on , however I don't see it appear in the  (also not when I'm not logged in).  Why is this. Aren't all bounty questions supposed to be visible there? Quantifiers "most" vs. "most of" I need to prove this equivalence   Let $a,b \in \mathbb{Q^{+}}$ then $\sqrt{a} + \sqrt{b} \in \mathbb{Q} $ if and only if $\sqrt{a} \in \mathbb{Q}$ and $\sqrt{b} \in \mathbb{Q}$  I have already proved the reciprocal (the obvious part) but I am having difficulties with the other part  Please any hint will be appreciated.  Thanks in advance. In  (scrolled to 5:55) there's a claim that replacing an AC high voltage transmission line with a high voltage DC transmission line allows for having seven times fewer powerlines (the video shows how seven lines evaporate, then one line appears and the cutting in the forest gets narrower).  How is that possible? I've read the  and it claims that     The power delivered in an AC system is defined by the root mean square (RMS) of an AC voltage, but RMS is only about 71% of the peak voltage. The peak voltage of AC determines the actual insulation thickness and conductor spacing.   but that doesn't explain seven times less power lines.  How is such massive (seven times) saving possible when switching from AC to DC transmission? I'd like to delete a file from my Time Machine backups, but there seems to be no option to do so in the Big Sur version of Time Machine. I go to the Time Machine menu and select &quot;Enter Time Machine&quot;. Then I see the view with the stack of windows. I browse to a file and select it. Now I want to delete it so it doesn't exist and take up space in backups. There is no option to delete it - either the action menu in the toolbar, or the control-click context menu. How do I delete a file from Time Machine backups? There used to be a way to do this. (I'd take a screenshot, but the key shortcuts for screenshots don't work in Time Machine. Not sure if this is just a Big Sur thing or if that's always the case.) Update: I should mention I'm using APFS for my Time Machine volume. That seems to be relevant. Also, I contacted Apple support about deleting entire snapshots (slightly different goal than the one here in this question). They said you can't do it anymore. It seems like there's some bugs here, because I have a &quot;Move to Trash&quot; option if I control-click one of my backups. It seems to work initially but then the backup comes back.  I don't like losing this much control over my files. I may look for another backup solution. Other than cost, is there a reason to not use higher rated components than called for by design? When drawing a line between two nodes in TikZ, you can specify the anchors on the beginning and end nodes in order to control how the line is drawn.  I would like to specify the anchor on the beginning node, and the angle at which the line is drawn, and let TikZ figure out where it will intercept the end node.  The usual use case is when I'm connecting to a "big node" and I would like the lines to be vertical.  I can cheat by placing a coordinate behind the big node, drawing a line to the coordinate instead, and then drawing the big node so that it covers things up.  However this won't work if I need to put an arrow on the end.  I could try to calculate the intercept coordinate on the edge of the big node manually, but that seems against the TikZ spirit.  |- doesn't work in this case.  \documentclass[10pt]{minimal} \usepackage[active,tightpage,pdftex]{preview} \usepackage{tikz}  \begin{document} \begin{preview}  \tikzstyle{RectObject}=[rectangle,fill=white,draw,line width=0.5mm] \tikzstyle{line}=[draw] \tikzstyle{arrow}=[draw, -latex]   \begin{tikzpicture}   %%%%%%%%%%%%%%%%%%%%%%%%   %%  LEFT SIDE   %%%%%%%%%%%%%%%%%%%%%%%%   \draw (1,0) node[RectObject] (Small1A) {A};   \draw (3,0) node[RectObject] (Small1B) {B};   \coordinate (Cheat1A) at (1,2);   \coordinate (Cheat1B) at (3,2);   \path [line] (Small1A.north) -- (Cheat1A);   \path [line] (Small1B.north) -- (Cheat1B);    % need to draw Big Node last to cover up our cheating   \draw (2,2) node[RectObject, inner xsep=1cm] (Big1) {Big Node};    \draw (2,4) node[text width=4cm] (text1) {L: Cheating to control line orientation. Arrows would be hard.};    %%%%%%%%%%%%%%%%%%%%%%%%   %%  RIGHT SIDE   %%%%%%%%%%%%%%%%%%%%%%%%   \draw (7,2) node[RectObject, inner xsep=1cm] (Big2) {Big Node};   \draw (6,0) node[RectObject] (Small2A) {A};   \draw (8,0) node[RectObject] (Small2B) {B};   \path [arrow] (Small2A) -- (Big2);   \path [arrow] (Small2B) -- (Big2);    \draw (7,4) node[text width=4cm] (text2) {R: No cheating. Arrows easy. Can't control line orientation.}; \end{tikzpicture} \end{preview} \end{document}   Here's a side-by-side example of "cheating", which shows what I want to be able to do, and "not cheating", which does not do what I want.    The left side of this example shows what I want to be able to do, except I want to do it without cheating, and I'd like to put arrows on the end of the lines. How do I get started creating an Ubuntu Desktop App? iBooks library like effect Prove Borel Measurable Set A with the following property has measure 0. This question is exercise 4.10 of Richard F. Bass's Real Analysis for Graduate Students, 2nd edition.  Let $\epsilon \in (0,1)$, let $m$ be Lebesgue measure, and suppose $A$ is Borel Measurable subset of $\mathbb R$. Prove that if $$ m(A\cap I)\leq (1-\epsilon)m(I)$$  for every interval $I$, then $m(A)=0$.  Collection of my thoughts  1. Try to prove by contradiction, suppose $m(A)=a&gt;0$ Borel Measurable Set satisfies the condition given. A property of any Borel Measurable Set on $\mathbb R$ is $A$ contains a closed set $F$ and is contained in open set $O$, and the measure of $F$ and $O$ can be arbitrarily close to $a$. 2. Obviously $A$ contains no interval, but it does not lead to contradiction, because we can find an example of $m(A)&gt;0$ that contains no interval(fat Cantor Set(remove middle 1/4 each time) has measure $1/2$ and contains no interval).  2'. For fat Cantor Set $C$ and given $\epsilon&gt;0$ how could we construct an interval I satisfies $m(C \cap I)&gt;(1-\epsilon)m(I) ?$   Maybe try to write Borel measurable set as countable union/intersection of intervals.   I'm stucked here and will appreciate any help, some examples/special case will be great too. It seems Microsoft has tried to solve this problem .  In Windows 7, there is no boundary between monitors and the mouse can move freely across the top of the screen between my 3 monitors, provided I had set their heights equal in display settings. Windows 8 introduced sticky corners. See .  Windows 10 got sticky corners as well. There are a few pixels at the top corners of each display where the mouse cannot cross over onto the other display. One must move the cursor down to avoid this region in order to get to the next display.  The image shows roughly the region where mouse movement is not permitted in Windows 10, but is allowed in Windows 7.    Personally, I had no problem with unrestricted mouse movement across the top of my screens - I got used to "aiming" for the x, and the convenience of unrestricted cursor movement. Like all the people who wanted to disable it in W8, I'm wondering if there is a way to disable it in W10.  Edit to address possible duplicate:  Although the problem is identical to the one in , solutions to solve the problem in W8 involving editing the registry key MouseCornerClipLength do not work in W10, since that registry key is not present in W10. Also adding that key and setting the value doesn't work. I searched the entire registry and couldn't find it in another location. No other W10 keys in the node referenced in the W8 solution are obvious replacements.  Edit to address possible solutions in comments  harrymc's  and  that worked for for Windows 8.1 do not work in Windows 10. How to format an inline source code Disable three-letter pings from chat to save Oliver, Olivier and Olive from Oli's pings Is there local file manipulation that's been done with JavaScript? I'm looking for a solution that can be accomplished with no install footprint like requiring .  Specifically, I'd like to read the contents from a file and write those contents to another file. At this point I'm not worried about gaining permissions and am just assuming I already have full permissions to these files. If a maximal subgroup is normal, it has prime index I'm having trouble with an exercise in "Introduce to the theory of group" book. This is problem:  Let $M$ be a maximal subgroup of $G$. Prove that if $M$ is a normal subgroup of $G$, then $[G: M]$ is finite and equal to a prime. Please don't throw this one out as a duplicate of  I'm not asking about the plurality of the noun immediately following those two words - I'm interested in exactly why the example below is problematic (context: I have several children, one named John)...     1: One of my children's name is John.   I can't see any obvious reason why singular name there is completely unacceptable, but it certainly doesn't sit well with me. It seems to turn on whether one of my children can be treated collectively as the "subject" of the Saxon genitive 's, but I don't see a problem with...     2: It's actually one of my children's, but you can sleep in this bed.   I realise that in practice we'd normally pluralise names in the first example. But of course, this implies that one of attaches to my children's names, which isn't really the same construction.  So - is there some kind of "rule, principle" debarring #1 above? Or is it just "one of those things"?    (Apologies if my later switch from brothers to children invalidates any comments or answers.) How can one express frequentist p-value in terms of Bayesian concepts? Let $b_n$ be a sequence which decreases monotonically to zero, prove $\sum b_nz^n$ converges for $|z|\leq 1$ and $z\neq 1$.  So I was able to prove it converges on the open unit disk in the complex plane, but except for a few special points on the boundary, such as $-1$ and $i$, where I was able to use the alternating series test, I can't seem to prove it converges in general on the boundary minus $z=1$.  I feel I have to somehow use the fact that all the numbers on the unit circle except $z=1$ rotate around and around as you raise them to higher and higher powers, thus you get the real and imaginary parts of your series to alternate between sets of positive and negative terms, but in a very irregular way for most initial angles.  I tried to simplify the problem by breaking the series into its real and imaginary parts, and then trying to prove they each converge separately, but I still can't prove either converges:  $$\sum_{n=0}^{\infty}b_ne^{n\theta i} = \sum_{n=0}^{\infty}b_n\cos(n\theta) +i\sum_{n=0}^{\infty}b_n\sin(n\theta).$$ Use LaTeX to produce Epub