Prove that $1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + ... + \frac{1}{\sqrt{n}}\geq \sqrt{n}$
Proving $\sum\limits_{k=1}^{n}{\frac{1}{\sqrt{k}}\ge\sqrt{n}}$ with induction
What constitutes a legal block in volleyball in regard to it not counting as the first hit? Today in a match the setter set the ball too low for me to spike it, so I just dinked it over.  The bottom of the ball was lower than the top of the net at the point where I made contact.  When the defender made contact, again, the bottom of the ball was lower than the top of the net (the top of the ball was over the top of the net though).  I felt that this wasn't technically a block and that he shouldn't have been allowed to hit it a second time, but I wasn't sure so I just let it go.  Was this a legal block or should it have been our point when he hit it a second time in a row?  Edit:  I don't think I was clear about the part where, after I hit the ball, the ball travelled over the net onto the other team's side of the court.  The other player did not reach over the net so there is no issue there.  It's just an issue of whether or not their first contact was a block or a hit.  I believe it was not a block but I can't find a specific ruling to address it. Should your PhD students call you by your first name? OAuth 2.0 access token request failed with error "must use HTTP POST" OAuth 2.0 Auth token request failed with below error   { "error": "invalid_request", "error_description": "must use HTTP POST" }   What does this error mean?  I am using Postman. Have a connected app. I have used Post method, Url has https. Same request works for other Orgs. Did I missed any configuration/Permission check?  https://testOrg.lightning.force.com/services/oauth2/token?grant_type=password&amp;client_id={{CLIENT_ID}}&amp;client_secret={{CLIENT_SECRET}}&amp;username={{USER_ID}}&amp;password={{PASSWORD}} Did all Headmasters get a portrait in Hogwarts? Story about a deadly alien creature on another planet that has some chemical in its body that makes it faster What functionality do 'marks' offer in the El Capitan Terminal? Since upgrading to OS X El Capitan, I've noticed a change in the Terminal: Executed lines show an opening bracket on the left, and a closing bracket on the right, as shown in this screenshot:    Similar questions have been asked here:          Some of the questions are referring to these brackets as marks, and the Terminal's Edit menu has a couple of entries related to marks, for example:   Edit > Marks > Mark as Prompt and Send Return Edit > Clear to Previous Mark Edit > Navigate > Jump to Previous Mark     Is there a summary of what these marks can be used for? I have not seen any reference to this new feature in any of the El Capitan feature overviews. I find in many places mentions of pre-amplifiers being used to boost audio signals prior to feeding to a power amplifier.  Wikipedia's article on  gives some hints at their usage, but it doesn't really explain why it would be necessary to have two amplifiers in series rather than a single one with greater gain.  If a pre-amp is just a power amp with a smaller gain, what's the point, and why the distinction between the two kinds? Is the distinction in function, or simply purpose? The startup disk cannot be partitioned or restored to a single partition. Windows 10 and Mojave The startup disk cannot be partitioned or restored to a single   partition.      The startup disk must be formatted as a single Mac OS Extended   (Journaled) volume or already partitioned by Boot Camp Assistant for   installing Windows.   I had installed Windows 10 via bootcamp in my mac. Current version is 10.14.1  Now I don't want windows and need that space back but getting above error when I go bootcamp to remove it.  This is the output of diskutil list  /dev/disk0 (internal, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *251.0 GB   disk0    1:                        EFI EFI                     209.7 MB   disk0s1    2:                 Apple_APFS Container disk1         181.3 GB   disk0s2    3:       Microsoft Basic Data BOOTCAMP                60.3 GB    disk0s3    4:           Windows Recovery                         509.6 MB   disk0s4  /dev/disk1 (synthesized):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      APFS Container Scheme -                      +181.3 GB   disk1                                  Physical Store disk0s2    1:                APFS Volume Macintosh HD            116.4 GB   disk1s1    2:                APFS Volume Preboot                 44.6 MB    disk1s2    3:                APFS Volume Recovery                512.8 MB   disk1s3    4:                APFS Volume VM                      4.3 GB     disk1s4   And this is of diskutil cs list  No CoreStorage logical volume groups found   Can anybody please help me to solve this problem?  Output after I tried commands given in the first answer  ❯ diskutil list /dev/disk0 (internal, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *251.0 GB   disk0    1:                        EFI EFI                     209.7 MB   disk0s1    2:                 Apple_APFS Container disk1         250.8 GB   disk0s2  /dev/disk1 (synthesized):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      APFS Container Scheme -                      +250.8 GB   disk1                                  Physical Store disk0s2    1:                APFS Volume Macintosh HD            116.4 GB   disk1s1    2:                APFS Volume Preboot                 44.6 MB    disk1s2    3:                APFS Volume Recovery                512.8 MB   disk1s3    4:                APFS Volume VM                      3.2 GB     disk1s4 I normally write using the American English forms, but for some particular words, I tend to naturally write it in the British manner (like with the word favourite rather than favorite).  Is it wrong to mix both styles? If not *wrong*, does it looks weird? Understanding the -exec option of `find` Bibliography in Texmaker It seems that  by the OP of  was attributed to the community user.   I'm assuming that the edit was by the OP from the edit summary:     updated my question   and from the actual update. Both texts are speaking in the first person. I want to write all zeros to my hard drive, so I can re-install windows 7 Will two-letter font style commands (\bf , \it , …) ever be resurrected in LaTeX? I understand that \bf and \it are now obsolete in LaTeX and that \textbf and \textit are proper, as they produce more sophisticated (in particular, cumulative and properly kerned) changes to font style. I have read the  and I understand the rules and their reasons, as well as the several other commands that are affected.  However, I find it convenient to use LaTeX for notetaking during lectures, and in that rushed environment, shortening a command by any number of keystrokes helps keep me from falling behind. \bf and its two-letter kin are still very useful to me for that reason, and once a presentation is finished I can go through and replace all appearances of \bf et al. with \textbf et al.  My question is this: is there a plan eventually to replace the short font style commands like \bf with the implementations of \textbf etc. some day, or should I expect \bf always to remain in existence but obsolete, for reasons of backward compatibility with original TeX? Original TeX has been greatly improved on in countless ways, but in the heat of transcription I sometimes miss its conciseness. Puzzle: Find largest rectangle (maximal rectangle problem) Automate keystrokes in Windows 7 64-bit I need to automate the pressing of the  key then pressing 5 digits then press  in Windows 7 64-bit.  I have an app that comes up on the screen and I need the "script" to press the keys.  I don't even need mouse support.  What can I use to do this?  Any examples or links? I have a function with the same name, but with different signature in a base and derived classes. When I am trying to use the base class's function in another class that inherits from the derived, I receive an error. See the following code:  class A {     public:     void foo(string s){}; };  class B : public A {     public:     int foo(int i){}; };  class C : public B {     public:     void bar()     {         string s;         foo(s);     } };   I receive the following error from the gcc compiler:  In member function `void C::bar()': no matching function for call to `C::foo(std::string&amp;)' candidates are: int B::foo(int)   If I remove int foo(int i){}; from class B, or if I rename it from foo1, everything works fine.   What's the problem with this? How to prove $\left\{ \omega|X(\omega)=Y(\omega)\right\} \in\mathcal{F}$ is measurable, if $X$ and $Y$ are measurable? Given a probability space $(\Omega ,\mathcal{F} ,\mu)$. Let $X$ and $Y$ be $\mathcal{F}$-measurable real valued random variables.  How would one proove that $\left\{ \omega|X(\omega)=Y(\omega)\right\} \in\mathcal{F}$ is measurable.  My thoughts: Since $X$ and $Y$ are measurable, it is true, that for each $x\in\mathbb{R}:$ $\left\{ \omega|X(\omega)&lt;x\right\} \in\mathcal{F}$ and $\left\{ \omega|Y(\omega)&lt;x\right\} \in\mathcal{F}$.  It follows that $\left\{ \omega|X(\omega)-Y(\omega)\leq x\right\} \in\mathcal{F}$  Therefore $\left\{ -\frac{1}{n}\leq\omega|X(\omega)-Y(\omega)\leq \frac{1}{n} \right\} \in\mathcal{F}$, for $n\in\mathbb{N}$.  Therefore $0=\bigcap_{n\in\mathbb{N}}\left\{ -\frac{1}{n}\leq\omega|X(\omega)-Y(\omega)\leq \frac{1}{n} \right\} \in\mathcal{F}$.  Am working towards the correct direction? I appreciate any constructive answer! Plurals of acronyms, letters, numbers — use an apostrophe or not? What are some examples of induction where the base case is difficult but the inductive step is trivial? According to :     ...proofs by mathematical induction   have two parts: the "base case" that   shows that the theorem is true for a   particular initial value such as n = 0   or n = 1 and then an inductive step   that shows that if the theorem is true   for a certain value of n, it is also   true for the value n + 1. The base   case is often trivial and is   identified as such, although there are   cases where the base case is difficult   but the inductive step is trivial.   What are some examples of proofs by induction where the base case is difficult but the inductive step is trivial?