This is what I have to prove:  $\lim_{n\rightarrow\infty}\inf a_n\leq\lim_{n\rightarrow\infty}\inf(\frac{1}{n}\sum_{i=1}^na_i)\leq\lim_{n\rightarrow\infty}\sup(\frac{1}{n}\sum_{i=1}^na_i)\leq\lim_{n\rightarrow\infty}\sup a_n$  Note, that $a_n$ and $b_n$ are bounded sequences. I'm gonna be totally honest: I have no clue where to even start with this proof. Can someone give me some advice?
This is a question from the book Methods of Real Analysis by R. R. Goldberg.     If $(s_n)$ is a sequence of real numbers and if $$\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$$ then prove that: $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$.   I don't have any idea how to start working on this problem. Please help. Thanks.
What are some good introductions to philosophy? I've been searching the questions posed here, and don't seem to find one that gives the answers I am looking for. I've proposed to start a  on meta but no one proposed anything, so I feel I must ask this question: What are some books generally regarded as good for a general introduction on philosophy? By this I mean good books that can introduce you to philosophy in general and throughout the times, not to some philosophy or period in particular.   almost answers mine, except for the fact that most things proposed there are not books, thus not giving me much options to weigh upon.  regards contemporary philosophy in particular. And  has the same problem as the first one, plus the fact that it asks for layman's terms, which is not what I am looking for.  I've been tempted to buy Will Durant's , but am not sure if it is the best book to get started on philosophy (plus, the book has some mixed reviews). I also checked , which seems to be unanimously good, but has the format of dictionary/encyclopaedia, which is not exactly what I am looking for (at least as a main source).  also seems to be a good introduction regarding how to do philosophy, rather than to its history and ideals (this also interests me, but I'd rather learn about the philosophers and their theories first).   It is also important to point out that I have little notions (or none at all) on philosophy. I would like to read something that covers from classical to contemporary philosophy generally (although I do not know if there is such a book), so that from there I could pick some author, period or branch of philosophy in particular to read further on. So, to sum up, I would like to 'hear' your opinions on what are some books generally regarded as good for a general introduction on philosophy. Segmentation fault error occurs for a simple function I have coded Why "hide" Luke from his father somewhere Anakin knows? In Episode II Anakin goes to the Lars homestead and meets his extended family. How would sending Luke there be "hidden" to Vader in any way? At least Leia had a random planet and family. Quantum entanglement for faster-than-light (FTL) network communication? Quantum entanglement is 2 atoms that are paired together and when you stop one from spinning the other also stops with the same spin. Can you use these pairs to have faster-than-light (FTL) communication between 2 computers? Would someone explain wheel sizes.  If I am not mistaken, the wheels on my Trek 520 touring bike are 700C. There are no markings, and of course, no documentation of any kind. I assume 700 means 700mm, so 700 / 25.4 = 27.56 inches.  If I measure the rim however, its diameter is approximately 25.1 inches.  If I buy Continental tires, the size is 28 inch, but I have to buy 27 inch Bontrager tubes.  To make matters worse, per this link, tires come in sizes of 26, 27, and 28 inch, and size 700.     How can I determine my wheel size, and are these various sizes interchangable. Also, is the 'C' suffix important, and what does it mean?  Of course the tires are marked, but I am under the impression (per the dealers comments) that Continental marks their tires differently than everyone else. Also, the tire markings don't help for ordering a new wheel.   With so many tire sizes available, how can I know if I am using the correct size.  After reading the various answers here, I should explain that my confusion on this matter stemmed from the fact that I used to work as an auto mechanic. In the auto industry, car tires are based on wheel size, 13", 14", 15" etc, and there is no possible way to mount a 15" tire on a 13" rim for example.  So when I saw all these different tire sizes, I assumed they all required a different wheel size, but in fact, there is essentially just one wheel diameter for my type of bike, which they refer to as 700C, which is actually 622mm.   I also initially assumed that the C referred to the wheel's width, but that isn't true either. The C is apparently just antiquated terminology.  So at least in theory, I can mount a 26", 27" 28" or 29" tire on my 700C (622mm) rim. I just need to verify the width of the tire is suitable my wheel width, which is what Sheldon Brown's article explains.  Also, I measured my Continental tires, which are marked as 28 inch. They actually measure 27 inches (fully inflated), which must be why my dealer calls their sizing non standard. A correct preposition - account closed due/by/at 12th of May I asked my bank to close my account on 12th of May. Also     I expect the account to be closed [?] 12th of May   . What preposition should be there? I think about by or due. Also if I want to ask someone a question: What date was your account closed [missing preposition?] Is there a specific term for the association of certain verb(s) with certain nouns (as direct objects)? Or vice versa. Or perhaps a name for the pair itself?  For example:   hold an opinion make a complaint offer an apology   Also, where might I find a list of such pairs?    EDIT: I chose my title wording randomly. I have no idea whether 'cognate verb' is the correct term for these things. However, I've just found out that 'cognate objects' are a defined class in linguistics, so perhaps my choice of words was inappropriate. I'm going to leave the title as it is, anyway. prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$ Work done against gravity The work done against gravity is $mgh$, well at least that's what my textbook says. I have a question: I can apply a force say 50N, so total work done = $mgh + mah$. Where $ma$ = Force. But the truth is irrespective of the force applied, the work done against gravity is always $mgh$. Why?  For example, when I move an object with a force, the work done is more, so work depends on the Force. But in case of gravity it always depends upon the weight I found  post about how to do Crow's Foot ERM Diagrams with Tikz, but it looks strange because the entity is outside the frame. However, it worked.  Then I found  even nicer version and tried to combine it, but sadly I'm not able to find out why it doesn't work. I'm not able to print two entities side by side and connect them with a crow's foot wire. They always print over each other...  How can I have both the lines and the style?  \documentclass{article}  \usepackage{tikz} \usetikzlibrary{shapes.multipart} \usetikzlibrary{matrix} \usetikzlibrary{positioning} \usetikzlibrary{shadows} \usetikzlibrary{calc}  \tikzset{     entity/.code={         \tikzset{             rounded corners,                          name=#1,             inner sep=2pt,             every entity/.try,         }%         \def\entityname{#1}%     },     entity anchor/.style={matrix anchor=#1},     every entity/.style={             draw,     },     every property/.style={         inner xsep=0.20cm, inner ysep=0.075cm, anchor=west, text width=1.75in     } } \def\property#1{\node[name=\entityname-#1, every property/.try]{\propertysplit#1;};} \def\properties{\begingroup\catcode`\_=11\relax\processproperties} \def\processproperties#1{\endgroup%     \gdef\propertycode{}%     \foreach \p in {#1}{%     \expandafter\expandafter\expandafter\gdef\expandafter\expandafter\expandafter\propertycode%             \expandafter\expandafter\expandafter{\expandafter\propertycode\expandafter\property\expandafter{\p}\\}%     }%     \propertycode% } \def\propertysplit#1:#2;{#1:\hfill#2}  \def\entitynamenode{% \node[every entity name/.try] (\entityname-name) {\entityname}; \draw (\entityname-name.south west) -- (\entityname-name.south east); \\[1ex] } \tikzset{   every entity name/.style={every property/.try, align=center} } \begin{document} \begin{tikzpicture}[every node/.style={font=\ttfamily}, node distance=0.5in] \matrix (m1) [entity=Employee] {     \entitynamenode     \properties{         Username:,         Password :  varchar(255),                 Name :  varchar(255),         LastName :  varchar(255)     } }; \matrix (m2) [entity=Student] {     \entitynamenode     \properties{         Username:,         Password :  varchar(255),                 Name :  varchar(255),         LastName :  varchar(255)     } }; \draw [-&gt;] (m1) -- (m2); % &lt;-- it break's here... %\draw [one to omany] (Employee-Username:) to node[above]{label} (Student-Username:); \end{tikzpicture} \end{document}      Edit: this is a Crow's Foot Notation for visualizing IT Database structures: Capturing a time in milliseconds Are there examples of third-(or higher)-order linear differential equations in physics or applied mathematics? The classical second-order linear ordinary differential equation is that named after Sturm and Liouville: formally, \begin{equation} (pu')'=ru. \end{equation} It arises naturally in many physical situations, for example through radial considerations of Schrödinger's equation. A higher-order "analogue" of it is the Orr$-$Sommerfeld equation, which, after relabelling the coefficients, may be written as \begin{equation} (\phi u'')''=\psi u. \end{equation} It also arises naturally, namely from certain simplifications applied to Navier$-$Stokes' (in)famous equation, and describes to great accuracy the cross-stream behaviour of channel fluid flow.  I have been thinking long and hard about whether there are additional higher-order linear differential equations that emerge naturally from our mathematical models of the world. It even seems pretty much all of the linear partial differential equations (heat, wave, Schrödinger, etc.) are of second order. Does anybody know of higher-order examples? SQL Server compatibility with New TLS Standards A word or short phrase for 'better in every way'? When we're comparing two or more things, there are times when we have to weigh the advantages or disadvantages of each one.  There are other times when one of the things is better for every way in which it's being judged, and the final decision is not based upon trade-offs.  I know that I've heard a phrase for this (I want to say it was two words), but I've had no luck in remembering it.    I don't think 'clearly superior' or 'clear winner' quite fit the bill, as that might still suggest that the disadvantages are significantly outweighed by the advantages.  Is there such a phrase, or even a single word, that would convey this meaning without using idioms? Sometimes I wonder if he is actually/really  as lavish as he   pretends to be.   I know that this sentence can work without either actually or really, but if we were to use one of them to make an impact on the listener, which one should it be? How do I determine which graphics card I'm using? I've lost my original power adapter, and I can't find the proper one in any of the stores currently available.  I've HP Compaq 6730b, and as per  it outputs 18.5V 3.5A. However, I've found an older power adapter I have which outputs 19.5V 4.62 A.  So, my question is: how much of a difference between those two it really is, and more importantly, how much of a risk would it be if I tried the power adapter I already have.  P.S. I know pretty much nothing about electricity/electronics. Theorem 3.2 Rudin -- How is 0 concluded? I'm having trouble understanding why the arbitrariness of $\epsilon$ allows us to conclude that $d(p,p')&lt;0$. It seems we could likely conclude a value such as $\frac {\epsilon}{100}$ couldn't we?? The other idea that would normally work is the limit (as $n$ approaches $\infty$, $p$ approaches $p'$) but that would mean we are further in another sequence which would have same problem Thanks in advance   $\ $  $\ $  $\ $   Definition of Convergence     A sequence $\{ p_n \}$ in a metric space $X$ is said to converge if there us a point $p \in X$ with the following property: For every $ \epsilon&gt;0$ there is an integer $N$ such that $n \ge N$ implies distance function $d(p_n, p) &lt; \epsilon.$ RSA, DES, AES, etc., all use (relatively) complicated mathematics to encrypt some message with some key. For each of these methods, there have been several documented vulnerabilities found over the years. Some even believe that there is a vulnerability in AES known only to the NSA.  On the other hand, the one-time pad is not only ridiculously simple, but also has been shown to be impossible to crack (if used appropriately).  Say you have some message to be encrypted with length n. Simply generate a random string with length m where m is at least as large as n, and perform modular addition of the random string onto the message.  As long as the random string is generated with a high-enough-quality random number generator, and as long as the same one-time pad isn't reused, it should be impossible to crack.  If this is true, all we would need to perfect are fast, cryptographically secure deterministic random number generators. Use the key as a starting seed for the RNG and you have a powerful uncrackable encryption scheme which is simple to understand and implement.  Blum-Blum-Shub is a fast, cryptographically secure PRNG, and there are others too.  I have written a C++ program which uses two 2048-bit primes (which match the necessary criteria) for Blum-Blum-Shub and uses a password which the user enters - converted into a starting seed - to run byte-wise modular addition. The process is fast, deterministic, and if the literature I have read is correct then it should be very secure.  So why is so much time and effort invested in coming up with convoluted encryption cyphers when mathematics shows that the simplest is often the best? Export each image from a collection in Google Earth Engine