Why None is the smallest in python?
Is everything greater than None?
Proving set theory question I need to prove the following letting A and B be subsets of a universal set U. $$A\cup B = U \iff A^c\subset B$$ It just isn't straight forward to me. Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer.  And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ diverges which also implies that the infinite series $\sum \limits_{k=0}^{\infty}a_k$ cannot be defined.  If I consider the infinite sum $1-1+1-1+1-1+\cdots$, then this statement is equivalent that I just defined the infinite series $\sum \limits_{k=0}^{\infty}a_k$, which is contradiction.  But, suppose that the sum $1-1+1-1+1-1+\cdots$ exists and let the value of the sum be $S$. Then, we can easily observe that $S=1-S$, therefore $S=1/2$. The supposition of this proposition already proved as false, but if I ignore the definition of infinite series, it holds.  Which one is right? Or are these propositions depends on what we define? Can I travel to Croatia with single-entry Schengen visa? I am traveling to Zurich with a single-entry Schengen visa.  While coming back can I travel to Croatia for 3 days and then return to India (Country of Origin)? I cannot remember anyone writing the letter epsilon in any other way than \varepsilon in any math class; but in LaTeX \epsilon and \varepsilon are different symbols. Do any of you know why there are two different symbols? (I.e. if \epsilon is the correct way to write the letter epsilon, why aren't mathematicians using it, and when is, according to the standards today, the correct situation to use each of the symbols?) I have latitude and longitude as 19.0649070739746 and 73.1308670043945 respectively.   In this case both coordinates are 13 decimal places long, but sometimes I also get coordinates which are 6 decimal places long.   Do fewer decimal points affect accuracy, and what does every digit after the decimal place signify? What is the structure in "as best you can"? I instinctively translate it "as best as you can", however this makes no sense.  What is the real structure behind this phrase?  I'll include an  to illustrate the historical presence of this phrase:    Whatever the structure is, I'm guessing that it has something to do with 19th century grammar. Fastest converging method for calculating e Showing a compact metric space has a countable dense subset I know that this is separable, but that has not been covered by the text at this point. I believe my proof (below) to be correct, but not very rigorous. Assuming that it actually is correct, could someone please help me make it more formal? (If it's not correct, please help with the proof). Thanks.  Proof: Let $X$ be a compact metric space. Let $B_n = \{B(x,\frac1{n}) : x\in X\}$. be the collection of open balls centered at $x$. Let $\{x\}^{(n)}$ be the set of centers for some particular $n$. Then by compactness of $X$, there are also a finite number of centers for a particular $n$ (corresponding to each open ball).   [This is where I have trouble properly saying what's on my mind:]  Consider $B(x,r_0)$, where $r_0 = \frac{1}{n_0}$. When we decrease the magnitude of $r_0$ to $r_1$, we must still be able to cover $X$. Meaning some open ball(s) must be "formed" where $B(x,r_0)$ was once "covering". So by adding open balls, we also add to our collection of centers. This means at $B(x,r_0)$, there was originally a point $y \in B(x,r_0)$ s.t. $B(y,r') \subset B(x,r_0)$ for some $r'$. That is, $B(x,r_0)$ contained another point of $X$. Thus, by definition, $X$ contains a countable dense subset. Prove that $\liminf x_n = -\limsup (-x_n)$ Set normal orientation of origin I want to change the orientation of my object to a specified direction.  Say I have an axle of a train whose normal orientation is set off by 90° from the global orientation:    The green arrow (Y-axis) points to the back of the train (along the global X-axis). But I want it to point to the sides of the train, so all I want is to rotate the orientation by 90° without rotating the object itself.  In other words, I want the normal and global orientation of the axle to be the identical:    Is this possible and if so, how?  Background: The game engine to use this model will move the bogies and the axles according to their orientations. After importing this train into the game the axles rotated 90° off. Nonabelian semidirect products of order $pq$? Best way to parse command line arguments in C#? When building console applications that take parameters, you can use the arguments passed to Main(string[] args).   In the past I've simply indexed/looped that array and done a few regular expressions to extract the values. However, when the commands get more complicated, the parsing can get pretty ugly.    So I'm interested in:   Libraries that you use Patterns that you use   Assume the commands always adhere to common standards such as . Run a program without administrative privileges in Windows 7 My account in Windows 7 is an administrator account. When I open the Run box (WinKey + R), I am warned, "This task will be created with administrative privileges."    Is there any way to run programs as myself, but without administrative privileges when logged in as an administrator? How do I resize/change resolution of pictures easily? How to edit 'etc/hosts' file in non-rooted phone? I'm using LineageOS 14.1 and don't want nor need to root it. How can I remount /system rw and edit /system/etc/hosts?  All answers in "" expect a rooted phone. How to test if a string is basically an integer in quotes using Ruby Symmetrical twin paradox in a closed universe Take the following gedankenexperiment in which two astronauts meet each other again and again in a perfectly symmetrical setting - a hyperspherical (3-manifold) universe in which the 3 dimensions are curved into the 4. dimension so that they can travel without acceleration in straight opposite directions and yet meet each other time after time.  On the one hand this situation is perfectly symmetrical - even in terms of homotopy and winding number. On the other hand the Lorentz invariance should break down according to , so that one frame is preferred - but which one?  So the question is: Who will be older? And why?  And even if there is one prefered inertial frame - the frame of the other astronaut should be identical with respect to all relevant parameters so that both get older at the same rate. Which again seems to be a violation of SRT in which the other twin seems to be getting older faster/slower...  How should one find out what the preferred frame is when everything is symmetrical - even in terms of GRT?  And if we are back to a situation with a preferred frame: what is the difference to the classical Galilean transform? Don't we get all the problems back that seemed to be solved by RT - e.g. the speed limit of light, because if there was a preferred frame you should be allowed to classically add velocities and therefore also get speeds bigger than c ?!? (I know  is only a local theory but I don't understand why the global preferred frame should not 'override' the local one).  Could anyone please enlighten me (please in a not too technical way because otherwise I wouldn't understand!)  EDIT Because there are still things that are unclear to me I posted a follow-up question: align xticks in PGFPlots How to suppress Terminated message after killing in bash? How can you suppress the Terminated message that comes up after you kill a process in a bash script?  I tried set +bm, but that doesn't work.  I know another solution involves calling exec 2&gt; /dev/null, but is that reliable? How do I reset it back so that I can continue to see stderr? List sorting by custom field I dont know how to sort a list of SObject Case.  I have a List&lt;Case&gt; newCases which i iterate over in for cycle, where i save Cases to my new List&lt;Case&gt; casesList.  for (Case newCase : newCases) {             if(newCase.Case_Content__c == '111' &amp;&amp; newCase.Status == '24'){                 String cuid = newCase.CUID__c;                 cuidSet.add(cuid);                 casesList.add(newCase);                 casesList.sort(); //TODO sorting by CUID__c             }         }   What i need to do is, to sort this new list by field CUID__c. So the Cases with same CUID are together.  Thank you for any help :)