Let $(X, d)$ be a metric space. Let $A, B ⊆ X$ be nonempty disjoint subsets and suppose $A$ is compact and $B$ is closed. Show that the sets $A$ and $B$ have a positive distance, i.e. show $d(A, B) := inf\{(a, b): a ∈ A, b ∈ B\} &gt; 0$
Claim: Let $X$ be a metric space. If $A,B\in X$ are disjoint, if A is compact, and if B is closed, then $\exists \delta&gt;0: |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$.  Proof. Assume the contrary. Let $\alpha_n\in A,\beta_n\in B$ be chosen such that $|\alpha_n-\beta_n|\rightarrow0$ as $n\rightarrow \infty$.  Since A is compact, there exists a convergent subsequence of $\alpha_n\;(n\in\mathbb{N})$, $\alpha_{n_m}\;(m\in\mathbb{N})$, which converges to $\alpha\in A$.  We have   $$|\alpha-\beta_{n_m}|\leq|\alpha-\alpha_{n_m}|+|\alpha_{n_m}-\beta_{n_m}|\rightarrow0 \;\;\;as\;\;m\rightarrow\infty.$$ Hence $\alpha$ is a limit point of B and since B is closed $\alpha\in B$, contradiction.  Is my proof correct? I feel as though I am missing something simple which would trivialize the proof.
Incubated but not hatched egg I got the message "oh" and it showed my egg. Before I could do anything the server went down. When I got back on my egg was gone. Is there anything I can do to retrieve my incubated but not hatched egg? \draw not working inside axis environment Judging from the pgfplots manual I expected the following code to work:  \documentclass{article} \usepackage{pgfplots} \begin{document} \begin{figure}   \begin{tikzpicture}     \begin{axis}[       xmin=0,       xmax=4,       ymin=0,       ymax=4,       ]       \draw (1,1) -- (3,2);     \end{axis}   \end{tikzpicture} \end{figure} \end{document}   However, it doesn't - meaning that it seems the \draw command does nothing at all. I just get an empty axes with no line.    Can anyone shed some light on this? I guess the solution will be really easy... Difference between staticmethod and classmethod What is the difference between a function decorated with  and one decorated with ? Android app intermittently shows blank body in questions and answers (multiple sites) Note: the bug has returned in version 1.0.89 in a somewhat different form. See:      This possible bug report is similar to the bug in  that was deemed not-reproducible.  I can reproduce something similar with more recent app &amp; Android versions.   App Version: 1.0.34 (original report; is still present in 1.0.38, 1.0.42, 1.0.58) Android Version: 4.4 (up to 4.4.4) Device: Motorola Droid Maxx   In the examples below, you can see some blank answer bodies.  These are on different sites, and represent answers with both edited and non-edited text. In the first example, I clicked on the top answer tab to get to the answers, and in the second example I clicked on the bottom "View n Answers" button. (Trying to be a good little tester!)  Because I've seen this blank-answer-body intermittently over the past couple weeks (before the May 7 update), my process to test was just to go to the first site I frequent and then systematically click through the question list until I found a question with answers in which at least one answer was blank.  For all three sites shown below, it was within the first 5 or 7 questions sorted by newest.  Please note that I have seen many questions with several answers, none of which were blank. It is truly intermittent.  Example blank answer body at Workplace.SE:    Example blank answer body at Programmers.SE:    Example blank answer body at UX.SE:    And now here's the kicker, which makes me think this is some sort of content-loading latency issue rather than a rendering issue:  I returned to the left-side menu, selected the User Experience site, picked the same question as above, and then clicked the top answer tab to view answers again, and lo and behold the text is there:    I then went back to the other questions with blank answers, shown above, and the answer bodies were no longer blank.  I am happy to go through more tests/paths and do screenshots if it is helpful.  The same bug also appears in question bodies. Installing Autodesk Maya on Ubuntu is possible, but can be a bit challenging.  Assuming that one already has the required specs and a solid graphics card driver installed, how can it be done? Simple programs with tutorials for beginners in graphic design? How to distribute objects in a circular pattern on a surface (Radial Array, Polar Array) long long implementation in 32 bit machine As per c99 standard, size of long long should be minimum 64 bits. How is this implemented in a 32 bit machine (eg. addition or multiplication of 2 long longs). Also, What is the equivalent of long long in C++. Sobolev inequality If $f\in H^2(\mathbb R^2)$, I want to show that    $||f||_{L^\infty}\le c||f||_{H^2}$ $||f||_{L^\infty}\le c||f||_{H^1} [1+\ln(1+||f||_{H^2})]$   For 1, I use $||f||_{L^\infty}\le \sup_{x\in \mathbb R^2} |f|\le\int_{\mathbb R^2} |\hat f(\xi)|d\xi = \int(1+|\xi|^2)^{-1}(1+|\xi|^2)|\hat f(\xi)|d\xi \le (\int (1+|\xi|^2)^{-2}d\xi)^2(\text {which is integrable in this case})||f||_{H^2}\to \text{is this correct?}$   But for 2, I am stucked. How can I get the "ln"? and how can I make it into a product of $H^1$ and $H^2$ norm? Does anyone know the general strategy for summing a series of the form:  $$\sum_{n=1}^{\infty} \frac{1}{n^2 + a^2},$$  where $a$ is a positive integer?  Any hints or ideas would be great! Can one executable be both a console and GUI application? I want to make a  program that can be run as a CLI or GUI application depending on what flags are passed into it. Can this be done?  I have found these related questions, but they don't exactly cover my situation: Is particle superposition reflected in the particle's gravitational footprint? NEWID() In Joined Virtual Table Causes Unintended Cross Apply Behavior My actual work query was an inner join, but this simple example with cross join seems to nearly always reproduce the problem.  SELECT * FROM (     SELECT 1 UNION ALL     SELECT 2 ) AA ( A ) CROSS JOIN (     SELECT NEWID() TEST_ID ) BB ( B )   With my inner join I had many rows for which I added to each a GUID using the NEWID() function, and for about 9 out of 10 such rows the multiplication with the 2-row virtual table produced the expected results, just 2 copies of the same GUID, while 1 out of 10 would produce different results. This was unexpected to say the least and gave me a really hard time trying to find this bug in my test data generation script.  If you take a look at the following queries using as well non-deterministic getdate and sysdatetime functions, you won't see this, I don't anyway-I always see the same datetime value in both final result rows.  SELECT * FROM (     SELECT 1 UNION ALL     SELECT 2 ) AA ( A ) CROSS JOIN (     SELECT GETDATE() TEST_ID ) BB ( B )  SELECT * FROM (     SELECT 1 UNION ALL     SELECT 2 ) AA ( A ) CROSS JOIN (     SELECT SYSDATETIME() TEST_ID ) BB ( B )   I'm currently using SQL Server 2008 and my work around for now is to load my rows with GUIDs into a table variable before finishing out my random data generation script. Once I have them as values in a table as opposed to virtual table, the problem goes away.  I have a workaround, but I'm looking for the ways to workaround without actual tables or table variables.  While writing this I tried without success these possibilities: 1) placing the newid() into a nested virtual table:  SELECT * FROM (     SELECT 1 UNION ALL     SELECT 2 ) AA ( A ) CROSS JOIN (     SELECT TEST_ID     FROM (         SELECT NEWID() TEST_ID     ) TT ) BB ( B )   2) wrapping the newid() within a cast expression such as:  SELECT CAST(NEWID() AS VARCHAR(100)) TEST_ID   3) reversing the order of appearance of the virtual tables within the join expression  SELECT * FROM (     SELECT NEWID() TEST_ID ) BB ( B ) CROSS JOIN (     SELECT 1 UNION ALL     SELECT 2 ) AA ( A )   4) using uncorrelated cross apply  SELECT * FROM (     SELECT NEWID() TEST_ID ) BB ( B ) CROSS APPLY (     SELECT 1 UNION ALL     SELECT 2 ) AA ( A )   Just before finally posting this question, now I tried this with success it seems, a correlated cross apply:  SELECT * FROM (     SELECT NEWID() TEST_ID ) BB ( B ) CROSS APPLY (     SELECT A     FROM (         SELECT 1 UNION ALL         SELECT 2     ) TT ( A )     WHERE BB.B IS NOT NULL ) AA ( A )   Anyone have any other more elegant, simpler workaround? I really don't want to use cross apply or correlation for a simple row multiplication if I don't have to. Do I need an airport transit visa for Dublin, Ireland en route to the UK? How to disable sticky corners in Windows 10 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. Cardinality of quotient ring $\mathbb{Z_6}[X]/(2X+4)$ Determine which circuits are parallel and which are in series I need to find the Equivalent Resistance across A and B. The problem is, I don't know which one's are in series and which ones are in parallel. How do I identify which ones are parallel or series? I've heard the number 7% thrown around a bunch of times, but I am struggling to find reputable data to back it up.   All I found was a quote from Warren Buffet that I couldn't find an original source for and data assembled by individuals on their personal websites. Is there something from an organization such as Market Watch making a statement about this?  I'm hoping to specifically find what the 10 or 20 year return is for investing in the S&amp;P 500 index fund or another diversified portfolio option. I've been riding a mountain bike for years, but I prefer to go longer distances on paved ground, so I think a road bike would be better. When I ask what is a good bike, I don't need brands or models - more like what components of the bike should I look for, and what should I avoid? Additionally, is it necessary to "go big or don't go at all" as far as spending - getting a real good quality road bike for $500 to a few thousand, or should I cheap out on a $100-200 road bike, as it is my first, then get a better one when I'm more experienced? Checking my understanding: $1 - 1 + 1 - 1 + 1 - ... = \frac{1}{2}$ I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...) = 1 - S$$ Therefore, $2S = 1$, so $S = \frac{1}{2}$. QED   I was under the impression that this sum doesn't converge, and therefore this step of the proof is invalid:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$   That is, it's not even valid to suppose that the sum is equal to some (implicitly real or complex) number $S$, and therefore the reasoning that follows is meaningless because $S$ doesn't exist and therefore has any properties we want it to have.  Is the reasoning I've given above correct? That is, is it correct for me to claim that the proof fails because $S$ doesn't exist?  (I've heard that there are techniques for evaluating divergent integrals by using complex analysis and Taylor series, so perhaps there's another way to prove that the summation is $\frac{1}{2}$; I just wanted to see whether my reasoning is sufficient to explain why this particular proof is incorrect. If the proof actually is correct, then I stand corrected!)  Thanks!