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Proofs with limit superior and limit inferior: $\liminf a_n \leq \limsup a_n$ I am stuck on proofs with subsequences. I do not really have a strategy or starting point with subsequences.   NOTE: subsequential limits are limits of subsequences  Prove: $a_n$ is bounded $\implies \liminf a_n \leq \limsup a_n$   Proof:  Let $a_n$ be a bounded sequence. That is, $\forall_n(a_n \leq A)$.  If $a_n$ converges then $\liminf a_n = \lim a_n = \limsup a_n$ and we are done.  Otherwise $a_n$ has a set of subsequential limits we need to show $\liminf a_n \leq \limsup a_n$:  This is where I am stuck... Need help with SQL Server PIVOT I know about the PIVOT feature in SQL Server and I've read a few other questions and answers on here but I just can't seem to grasp it in order to do what I need. I'm trying to run some calculations and I have the following:  TABLE office_settings column office_id (bigint not null) column setting_value (NVARCHAR(500) NOT NULL)  office_id    setting_value -------------------------------------------- 4             0.2 4             0.2 4             2 4             3 5             0.2 5             0.2 5             2 5             3   I'd like to convert this using PIVOT (assuming that's the right tool for the job) to this, basically "grouping by" the office_id column, so:  office_id    value_1    value_2    value_3   value_4 ------------------------------------------------------     4         0.2         0.2         2         3     5         0.2         0.2         2         3   I should also mention that I need to CAST the setting_value column values to DECIMAL(4,2) in the process of this - in case that has any bearing on the T-SQL required to accomplish it.  Any ideas? How to repair a broken .EXE file association Is there a word for colloquial forms of address? For example, "dude," "man," "buddy," "pal," etc, when used to stand in for someone's name.  "Hey, pal, how's it going?"  Is there a word for terms like these?  Or is "colloquialism" as close as we can get? I've never really got how chmod worked up until today. I followed a tutorial that explained a big deal to me.   For example, I've read that you've got three different permission groups:   owner (u) group (g) everyone (o)   Based on these three groups, I now know that:   If the file is owned by the user, the user permissions determine the access. If the group of the file is the same as the user's group, the group permission determine the access. If the user is not the file owner, and is not in the group, then the other permission is used.   I've also learned that you've got the following permissions:   read (r) write (w) execute (x)   I created a directory to test my newly acquired knowledge:  mkdir test   Then I did some tests:  chmod u+rwx test/ # drwx------ chmod g+rx test/ # drwxr-x--- chmod u-x test/ # drw-r-x---   After fooling around for some time I think I finally got the hang of chmod and the way you set permission using this command.    But...  I still have a few questions:   What does the d at the start stand for? What's the name and use of the containing slot and what other values can it hold? How can I set and unset it? What is the value for this d? (As you only have 7=4+2+1 7=4+2+1 7=4+2+1) Why do people sometimes use 0777 instead of 777 to set their permissions?   But as I shouldn't be asking multiple questions, I'll try to ask it in one question.  In UNIX based system such as all Linux distributions, concerning the permissions, what does the first part (d) stand for and what's the use for this part of the permissions? Proving $\frac{1}{n+1} + \frac{1}{n+2}+\cdots+\frac{1}{2n} &gt; \frac{13}{24}$ for $n&gt;1,n\in\Bbb N$  To solve it I used induction but it is leading me nowhere my attempt was as follows:  Lets assume the inequality is true for $n = k$ then we need to prove that it is true for $k+1$  so we need to prove $\frac1{k+2} + \frac1{k+3}+\cdots+\frac1{2(k+1)} &gt; 13/24$  I don't know where to go from here please help. My recollection is very sketchy but this is what I remember....   Film was possibly black and white I saw it as a child in the late '80s early '90s but it's older It was subtitled and in Russian or similar language It involved some astronauts  travelling in space with possibly time travel The astronauts (mostly men) encounter a planet/civilization of only women The women no longer need men to breed and are so become superior and possibly want to eliminate all men    That's all I have to go on, sorry. How do I tell if ACR will work with my camera and how do I get the correct version for Photoshop? I am a beginner photographer with a Canon 7d and MacBook Pro. I'm not sure if Adobe camera Raw is compatible with either and really need advice on how to go about downloading the correct version. How to extract logs between two time stamps I want to extract all logs between two timestamps. Some lines may not have the timestamp, but I want those lines also. In short, I want every line that falls under two time stamps. My log structure looks like:  [2014-04-07 23:59:58] CheckForCallAction [ERROR] Exception caught in +CheckForCallAction :: null --Checking user-- Post [2014-04-08 00:00:03] MobileAppRequestFilter [DEBUG] Action requested checkforcall   Suppose I want to extract everything between 2014-04-07 23:00 and 2014-04-08 02:00.  Please note the start time stamp or end time stamp may not be there in the log, but I want every line between these two time stamps. Mathematical journal for an undergraduate I want to make someone a present by subscribing for him to a mathematical journal. (It should appear every month or every two months). The contents should be comprehensible to an undergraduate in mathematics, and be about a broad selection of topics (for example not only analysis).   I expect the content to be something like proposing interesting problems and then giving solutions with some references for further study. Are there some famous journals you could suggest me? show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$ On the metric subspace $S = [0,1]$ of the Euclidean space $\mathbb R^1 $, every interval of the form $A = [0,a)$ or $(a, 1]$ where $0&lt;a&lt;1$ is open set in S. These sets are not open in $\mathbb R^1$   Here's what I attempted to show that $A$ is open in $S$. I have no idea how it is not open in $\mathbb R^1$.  Let $M = \mathbb R^1$, $x \in A  = [0,a)$ . If $x = 0$, $ r \leq \min \{a, 1-a\}, \\\  B_S(0; r) = B_M(0,r)\cap[0,1] = (-r, r) \cap[0,1] = [0,r) \subseteq A $  If $x \neq 0, r \leq \min \{x,|a-x|, 1-x\},  \\B_S(x; r) = B_M(x,r)\cap[0,1] = (x-r, x+r) \cap[0,1] \subset (0,x) \text{ or } (x, 1) \subset A$ How to stop the find command after second match? How to make find command, that will show only first two matches and then end, something similiar to  just version for two records. Fastest way to convert string to integer in PHP Using PHP, what's the fastest way to convert a string like this: "123" to an integer?  Why is that particular method the fastest? What happens if it gets unexpected input, such as "hello" or an array? Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$ How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$?  I ran into this identity on . I haven't been able to find any proof online and my efforts to get from $\ln(x) := \int_1^x \frac{\mathrm dt}t$ to that limit have been a failure. Is there an optimal loss function for dealing with imbalanced classes? I'm aware that there are many ways of dealing with datasets where there is a strong class imbalance in the target variable: downsampling the more prevalent and less important class, over-weighting the valuable minority class, etc. Are there any loss functions that are ideally suited for these types of problems? I've heard of AUC, but AUC sounds like a means of evaluating a model once its training is complete (correct me if I'm wrong). Log loss and squared loss penalize a model's over confidence, but overconfidence and class imbalance seem unrelated. Is that true? When I try to upload an image, I'm presented with the following dialog box,    The dialog box does not allow uploading either from my computer or by posting a link.  This happens only on Chrome browser (running on Windows). Clearing cache and cookies, and restarting computer do not resolve.  Is this just my computer? Or is this a site-wide bug?  Additional info:  Clicking on 'click' to upload does nothing. Dragging an image onto dialog redirects me to the file location on my computer. Clicking 'Add picture' redirects me to http://[SE-domain].com/upload/image?https=true.  UPDATE: A Norton extension for Chrome was the source of the problem. Pgfplots: how to fill bounded area under a curve using addplot and fill? Constructing the reciprocal of a segment How can one construct the reciprocal length of a line segment? For example, given any line segment a, how can $\frac{1}{a}$ be constructed?  I was told that it can be solved by creating similar triangles, but I do not get it. I have made a fancy title page using titleformat and tikz, but I now face the following problem. The \chaptertitlename command returns \chaptername or \appendixname. I would like to spread the letters of the text returned by \chaptertitlename to a given length.  For example I would like to be able to call: \mycommand{\chaptertitlename}{3cm} in order to space the letters of \chaptertitlename on a total length of 3cm. How to do that ?  EDIT: This example illustrates a nearly working thing:  \documentclass[11pt]{article} \usepackage{fontspec} \newcommand{\spreadletters}[2]{\makebox[#2][s]{#1}}  \begin{document} \spreadletters{m y w o r d}{3cm} \end{document}   How to convert myword to m y w o r d ?  EDIT 2: Can someone explain me why the first work and not the second one (and how to make it work) ?  \documentclass[11pt]{book} \usepackage{fontspec} \usepackage{seqsplit} \usepackage[explicit]{titlesec}  \newcommand{\spreadletters}[2]{\makebox[#2][s]{\seqsplit{#1}}}  \begin{document} \chapter{First} \spreadletters{myword}{10cm} \linebreak % This is working \spreadletters{\chaptertitlename}{10cm} % This is not working \end{document} Prove linear combinations of logarithms of primes over $\mathbb{Q}$ is independent Suppose we have a set of primes $p_1,\dots,p_t$. Prove that  $\log p_1,\dots,\log p_t$ is linear independent over $\mathbb{Q}$. Now, this implies $ \sum_{j=1}^{t}x_j\log(p_j)=0 \iff x_1=\dots=x_t=0$.  I think I have to use that fact that every $q\in\mathbb{Q}$ can be written as $\prod_{\mathcal{P}}$, where $n_p$ is a unique sequence ($n_2$,$n_3$,$\dots$) with domain $\mathbb{Z}$. Here, $\mathcal{P}$ denotes the set of all integers.  Now how can I use this to prove the linear independency?