Proving that the average approaches zero If the positive numbers $v_n$ approach zero as $n\to\infty$, prove   that their average $(v_1+v_2+\dots+v_n)/n$ also approaches zero.   We have:  For any $\epsilon$, there is an $N$ such that $v_n&lt;\epsilon$ if $n&gt;N$.  We need to prove:  For any $\epsilon$, there is an $N$ such that $\frac{v_1+v_2+\dots+v_n}n&lt;\epsilon$ if $n&gt;N$.  I am having a hard time connecting these two statements. I have tried with examples but I have not been able to get through these symbols. Intuitively, average follows instantaneous. $\bar v$ rises and falls with the next $v$. But I need a formal proof.
How to prove that if $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$ I'm wondering if I have a sufficient proof of the following:   If $(a_n)$ is a sequence such that $\lim_{n \rightarrow \infty}a_n=A$, then $\lim_{n \rightarrow \infty}\frac{a_1+...+a_n}{n}=A$.   My approach:  For all $\varepsilon &gt; 0$, there exists $N$ such that for all $k&gt;N$, $|a_k -A|&lt;\varepsilon$. So we can break the limit up as follows $$\lim_{n\rightarrow \infty} \frac{a_1+...+a_k}{n} + \lim_{n \rightarrow \infty}\frac{a_{k+1}+...+a_n}{n} \overset{\epsilon \rightarrow 0}{=} 0 + \lim_{n \rightarrow \infty}\frac{nA}{n}=A$$  Is this on the right track, or am I missing something about breaking up the limit in the way I have?
What is the name of the genre of games similar to Jetpack Joyride? Seeking neighborhood boundaries globally? Besides Zillow's creative-commons licensed  shapefiles, what sources of neighborhood boundaries are there? Not just in the US, but all over the world.  Free/open source is ideal, but paid sources could be interesting too. How do you prove that $e=\sum_{n=0}^{\infty}\frac{1}{n!}$? Here I am assuming $e:=\lim_{n\to\infty}(1+\frac{1}{n})^n$. Do you have any good PDF file or booklet available online on this? I do not like how my analysis text handles this... PythonTEX not working on current Overleaf compiler Is there a way to make PythonTEX compile as of today in Overleaf V2 ?  \documentclass[]{article} \usepackage[utf8]{inputenc} \usepackage{pythontex}  \begin{document} \begin{pyverbatim} def calc_ellipticity(theta, delta):     theta = np.deg2rad(theta)     psi = np.arcsin(np.sin(2*theta) * np.sin(delta))/2     return np.tan(psi) \end{pyverbatim} \end{document}   Compiling gives : ? PythonTEX ? On my system (Windows Vista Home Premium 64bit) Python files are associated with Python interpreter:  C:\Users\Piotr&gt;assoc .py .py=Python.File C:\Users\Piotr&gt;ftype Python.File Python.File=c:\python\2.7\python.exe "%1" %*   Nevertheless when I run any Python script from command line a window titled Open With pops up asking me to Choose the program you want to use to open this file. How to insert strings containing slashes with sed? I think the title is pretty self explanatory.   The question is how can I create an alias in a Windows cmd that will also work after I close and then reopen it. For example I would like an alias called ip to call the command ipconfig or an alias ls -l to call Dir.  My operating system is Windows 10, in case that makes a difference. Let $X, Y$ be an exponentially distributed random variables with parameters $a, b$. Then $X$ has pdf:    $$f_X(x) =\begin{cases} a e^{-a x},&amp; x\geq 0\\ 0,&amp; \text{otherwise}.\end{cases}$$ Suppose $X$ and $Y$ independent. Show that $$\mathbb P(X&gt;Y) = \frac{b}{a+b}.$$   Now I thought the following: $$f(x,y) = f_X(x)\ f_Y(y) = abe^{-ax -by},\qquad\text{for } x,y &gt; 0.$$ And then $$\mathbb P(X&gt;Y) = \int_0^\infty \int_0^x a b e^{-ax -by}\,dydx$$ However, if I solve this (manually or using Wolframalpha), I can't seem to end up with $\frac{b}{a+b}$. Any ideas? Is it "I" or "me" in "Keep Tom and I/me updated"? Can you use Uncanny Dodge against extra damage effects? Showing $\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_0^1 \mid F^{-1}(u)-G^{-1}(u)\mid du$ with $F$, $G$ CDF functions Let's $X$ and $Y$ have CDF functions admitting moment of order $1$. Let's be $F$ cdf of $X$ and $G$ cdf of $Y$.  I want to show that $$\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_{0}^{1} \mid F^{-1}(u)-G^{-1}(u)\mid du\,.$$ Facing an error "*** glibc detected *** free(): invalid next size (fast)" Is there a general formula for finding all subgroups of dihedral groups? table of contents no word separation Is it possible to have in the toc no word separation but have a word separation in the chaptername:  For example:   Contents  1. max Mustermann   Chapter  Max Muster- mann I don't have cd burner, so I want to install W7 on my 2nd computer via an USB-Drive. I just copied the files from the iso but it won't boot. So I installed ms-sys and did a ms-sys -7 /dev/sdb but it just says "Boot error".  How can I install it the right way? Tried many different tutorials by now but none worked. What is the difference between re.search and re.match? What is the difference between the search() and match() functions in the ?  I've read the  (), but I never seem to remember it.  I keep having to look it up and re-learn it.  I'm hoping that someone will answer it clearly with examples so that (perhaps) it will stick in my head.  Or at least I'll have a better place to return with my question and it will take less time to re-learn it. LaTeX: optional arguments with square brackets LaTeX is trying to add a missing $ that I don't want to the following code. Minimal working example:  \documentclass[letter,12pt]{article}  \newcommand{\monkeyBeans}[2][]{   \parbox{4in}{#1} }  \begin{document}  \parbox{4in}{$[a, b]$}   \monkeyBeans[$[a, b]$]{yo monkey}  \end{document}   If I try any of the following lines in my document instead of the call to monkeyBeans, there is no error:  \parbox{4in}{$[a, b]$}  \monkeyBeans[yo monkey]{$[a, b]$} \monkeyBeans[$(a, b)$]{yo monkey} \monkeyBeans{$[a, b]$}{yo monkey}   What is the deal with square brackets in optional arguments? Why my DCIM folder is empty seen in the computer but not empty in the phone I am using HuaWeiNova device, and I tried to access it with aft-mtp-cli. $ aft-mtp-cli $ ls DCIM/  I got folders listed here: |-photos |-Camera ....  When I tried to list or download my photos in the Camera/ folder, I got nothing. Only the empty folders are listed and the folders that contains files are not shown at all. I am quite sure my photos are stored in the Camera folder in my cell phone. How come I could not access my photos with my computer ? Old fantasy book with a party finding coloured stones having powers over fire, water, mood I'm trying to remember the name of a book series of which I remember only the following, with Google yielding no results.   the name of the author starts with a Z   the main character is a knight in an household with a name sounding like Malachai who goes on a quest with six others guys and girls (a bard, another knight, a priest, etc.) and each of them finds a stone of colour which has a power on various things (like fire, their mood or water) and then they set toward a quest to beat some ancient evil. I used to find the book at the local library 10 years ago in France (but it was translated). Add a rule after chapter title using titlesec