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How to prove a total order has a unique minimal element Let $R$ be a total order on set $S$. Prove that if $S$ has a minimal element, than the minimum element is unique.  I have difficulties with proofs. I know any graph of a total order is a straight line, which clearly has a minimal element. How doI tell when something is mathematically proven?  EDIT: this is how I would prove it  Let $x, y \in S$ such that x and y are minimal. Since S is a total order $x$ and $y$ are comparable and either    case 1: one comes after the other and therefore one isn't the minimal case 2: both are the same and $x=y$  I know this isn't a mathy argument and could use help making it more presentable. What spell slot count formula can I use to base my homebrew mechanic on? Add a message when posting to old questions with more than N answers I've been trying to photograph lightning lately, as we have had some great storms in Colorado. I've had some difficulty getting a good vantage point, however, as almost all of these storms include rain as well. To keep my camera from getting drenched (I have weather-sealed lenses, but my camera body, a Canon 450D, is not weather sealed as far as I can tell), I've had to shoot under the eve of a building or under a thick canopy of trees, however that has greatly limited my ability to adapt to the changing weather and get get a vantage point on the best lightning.  Is there any equipment, something I can purchase or something home-made that has easy-to-find parts, that can help me protect my lens from the rain while still being able to shoot? Additionally, I've been getting rain on a UV filter I have on the front of my lenses, simply to prevent rain from getting on the front lens element itself. However, once a storm really gets going, so does the rain. Is there a simple solution to keep rain off the front of the lens? I have watched the series of Two and a Half Men two times now till the latest episode, but can't understand why either Charlie or Walden just don't kick Alan out? I mean it is a bit cruel but they are shown to put up with him even though he gets up on their nerves (even got on my nerves).    Some examples off the top of my head:   He became the reason of his first engagement with Mia to be broken up.   He once set up his curtains on fire after ruining his day.   He invited his girlfriend/assistant to move in without his permission and started acting like he owned part of the house when he didn't even pay rent. He acted sickly and dying (Walden's time) and tricked him out of money for his care and comfort.   He deliberately tried to break his relations (Walden's) just so he can keep sponging off.   I know not the best examples, but if not in every episode then in every alternate episode he is shown to getting up on their nerves and they just keep on telling him to leave. Why don't they just kick him out? Why put up with such a leach? Using same modulus for RSA How to select records with maximum values in two columns? Plot a data/Table from Mathematica using `pgfplots` Can Mjolnir "stick" to anything? Re-entering the US for the third time in a year with total of 240 Days I went to the US for 90 days on an ESTA in August to November. Then back to my home country for 1 month and then back again mid December until mid February and now I am going back again after 10 days for another 90 days. Now I can see that some people refer to 180 day rule in a year. My stay will be 240 days in one year with short gaps between coming back. Is the 180 day rule accurate? Some people write that you can do unlimited trips to The US as long as you don't stay more than 90 days. Is that correct? My reason for going there the first time was to visit and travel with an American friend.  After my first visit (she is my girlfriend) moved with me to my home country but has ended up in a custody battle in the US so she needed to return urgently so I returned only to be with her and support her, not to be a tourist.  Then after my second visit I needed to go back to my sick Dad and to rent out my apartment since we left in a rush to go back to the US.  Now, my third visit is to continue to be with my girlfriend and support her in her difficult situation.   My question is, if I do end up in trouble should I tell them this whole story? Or should I just say I am visiting a friend for a third time because I want to see more of the country and I have some more time off before I go back to my job.  I have proof of money in my bank account and I stay and eat with her for free Documented proof that I have a job to return to in my home country My return ticket printed and an address to stay at in the US and I have proof I have an apartment in my home country Evaluating the sum after reordering an infinite series $1-\frac {1} {2}+\frac {1} {3}-\frac {1} {4}+\ldots $ If in the series  $1-\dfrac {1} {2}+\dfrac {1} {3}-\dfrac {1} {4}+\ldots $ the order of the terms be altered, so that the ratio of the number of positive terms to the number of negative terms in the first $n$ terms is ultimately $a^{2}$, show that the sum of the series will become $\log \left( 2a\right) $.  Solution attempt. Let $p$ be the number of positive terms in the first $n$ terms of the reordered series so based on the question we are allowed $a^{2}=\dfrac {p} {n-p}$. Now solving for p we get $p=\dfrac {a^{2}n} {\left( 1+a^{2}\right) }$ and $n-p=\dfrac {n } {\left( 1+a^{2}\right) }$.  We also observe that in the original series only the odd terms are positive and only the even terms are negative.  Let's define $S_{odd}=1+\dfrac {1} {3}+\dfrac {1} {5}+.\ldots +\dfrac {1} {2n-1}$ and $S_{even}=\dfrac {-1} {2}\dfrac {-1} {4}\ldots -\dfrac {1} {2n}$  $S_{Reordered}=S_{odd_{p}}+S_{even_{n-p}}$  $S_{Reordered}=\sum _{t=1}^{t=P}\dfrac {1} {2t-1}+\sum _{t=1}^{t=(n-P)}\dfrac {-1} {2t}$ $S_{Reordered}=\sum _{t=1}^{t=\dfrac {a^{2}n} {\left( 1+a^{2}\right) }}\dfrac {1} {2t-1}+\sum _{t=1}^{t=\dfrac {n } {\left( 1+a^{2}\right) }}\dfrac {-1} {2t}$    In order to extend $S_{Reordered}$ from n terms to an infinite length. I guess i should take the limit of the $S_{Reordered}$ as $n\rightarrow \infty $ which would make the upper values of $t$ (Not sure of technical term here) to be $\infty$ giving me   $S_{Reordered}=\sum _{t=1}^{t=\infty}\dfrac {1} {2t-1}+\sum _{t=1}^{t=\infty}\dfrac {-1} {2t}$   and i have lost the $a$ from the expression. I guess i am stuck i need to perform some step to capture a before taking the limit. Any help would be much appreciated. Commutator Subgroup and the Largest Abelian Quotient Good day. I'm not too sure how I should go about asking my question, so if it is in anyway confusing, feel free to edit it.   In my Group Theory course we defined and showed a few properties of the commutator subgroup $[G,G]$ of a group $G$. And then we stated and proved a lemma in lecture that I'm having some difficulty wrapping my head around. I'll state it here for clarity.  Lemma a) The commutator subgroup $[G,G]$ of $G$ is a normal subgroup of $G$. b) The quotient group $G / [G,G]$ is the largest abelian quotient of G in the sense that every homomorphism $\varphi : G \to A$ where $A$ is an an abelian group, induces a unique homomorphism $\bar \varphi : G/[G,G] \to A$ so that $\varphi =\bar \varphi \circ \pi$ where $\pi : G \to G/[G,G]$ is the quotient homomorphism.  Now, it's not the proof of this lemma that I'm having trouble with, it's the italicized text that is messing with me. I completely understand what it is that the italicized text is saying, I'm just not too sure why it is that the property gives us the "largeness" that is claimed. I can see that the above forces $G$ to "pass through" $G/[G,G]$ before it "reaches" $A$ (I apologize for the casual terminology), so I can sort of see why $G/[G,G]$ should be the largest quotient, in a non-rigorous sort of way.  Overall I'm very confused. Perhaps I'm missing something very simple. Any help or nudges in the right direction would be much appreciated.   As a side note I did try a search on Google and here on Stack Exchange and couldn't find anything that answered my question. So if I missed something obvious, then my apologies. Thank you in advance. I'm new to Blender and this site, so I apologize in advance if I'm not following any of the proper etiquette.  I'm no artist, but I'm trying to learn Blender so that I have some programmer art for me to work around in Unity.    I followed a tutorial for making a basic human model, I rigged it with rigify, and now I'm starting to export it following yet another tutorial.  The thing is, when I was trying to rig the model, I couldn't get the model to go transparent so that I could see the bones inside the model.  Eventually, I managed to get it into wireframe mode, but I can't get it back into solid mode.    It's probably just a display thing and not a property of the model itself, but I want to be sure.  I'm looking at the display options under the "Figure" pane in Blender, and none of the Display check boxes are checked, including the one that says "Wire".  I don't want to import a plain wireframe model into Unity, and even if that doesn't happen, this is functionality of Blender that I should really know anyway, so I'd appreciate any help that anyone has to offer.  If necessary, I can upload the .blend file, but I'm not sure what site people would prefer that I upload it to. is there a way to request a new tag be created? Possible Duplicate:          I do not have sufficient reputation on any site to create new tags, but I do on occasion find myself in need of a tag that does not exists.  Is there a feature that allows for a person with sub-1500 reputation to request a new tag? I want to start using the terminal more often, but I don't know what are the different commands available to me. Is there a way to list all the different commands that I can make use of? Shapefile layer no longer editable (toggle edit option greyed out) in QGIS We were asked to prove the following:  Let $ A $ be an $n \times n$ matrix with $n$ distinct real eigenvalues. If $AB=BA$, show that $B$ is diagonalizable.   It was suggested I show that an eigenvector of $A$ is also an eigenvector of $B$. I am both having trouble doing this and failing to see how I would complete the proof after. Any help would be appreciated.  Thanks Explanation for "emails"? PlotRange precision limit? Completely clear all output from a Terminal window