Let  $$ \Delta P = (b-P_n)/2 \\ P_{n+1} = P_n + \Delta P\\ P_n : \mathbb{Q} \to \mathbb{R} $$ We shall attempt to prove that  $$ \lim_{n \to \infty} P_n = b $$ I'll try in three parts   $P_n$ is monotonic $P_n$ is bounded by $b$ Something along the lines of $\sup P_n = b$   First, prove $P_n$ is bounded by $b$. Assume the contrary by taking $P_n$ to be the supremum of $P_n &lt; b$ and therefore $P_{n+1} &gt; b$. Now that implies $\Delta P &gt; b-P_n$ which violates the definition of the recurrence. Therefore, $P_n$ is bounded from above by $b$  Second, let us prove that $P_n$ is monotonic with respect to $n$. Assuming $b &gt; P_0$, $P_1 &gt; P_0$. Now choose an arbitrary $P_{n}$, where $b &gt; P_n$ from the previous result. If $P_{n+1}  = P_n + \Delta P$, $\Delta P = \frac{b-P_n}{2} &gt; 0$ and therefore $P_n$ is increasing for $P_n &lt; b$  Now let us prove that $\lim_{n \to \infty} P_n = b$.  That means that for all $M, \epsilon &gt; 0$  $$ n &gt; M \implies |P_n - L| &lt; \epsilon $$ To start, we know $L \le b$ by our first result.   Now all we must prove is that $L &lt; b$ is impossible, which is what I am having trouble with.   What I am having trouble with  Well if for all $n&gt;M$, $P_n &lt; L$ then for all $P_n$, $$ \Delta P  =  \frac{b-P_n}{2}&lt; L-P_n &lt; \epsilon $$ I'm having trouble constructing a contradiction and was hoping for some help doing so.   I specifically want to use a delta epsilon proof and not any other form of proof.
I'm taking Calculus II next semester so I don't have all the tools from it to back me up but something came up in a discussion this evening that asked me to prove that the following sequence converges. As I'm new to formal math I'm hoping someone can verify my proof, especially step $6$. I must prove that $$\lim_{n\to\infty}P_n=b$$ where $$P_{n+1} = P_n + \frac{b-P_n}{2}$$ $$P_0 = a$$ and $a,b \in \mathbb{R}$ such that $a&gt;b$ My proof goes as follows:   We must prove that $$\lim_{n\to\infty}P_n=b$$  That is, $\forall\epsilon&gt;0$ there exists a $\delta$ such that there exists an $n$ satisfying $$n&gt;\delta \iff b-P_n&lt;\epsilon$$  First, let us prove that $P_n$ is bounded by $b$ by contradiction. Let us choose an $i$ such that $P_i &lt; b$ and $P_{i+1}&gt;b$. Then we have $P_{i+1} = P_i + \Delta P$ where $\Delta P &gt; b-P_i$. but this is impossible because by the definition of the recurrence $\Delta P = \frac{b-P_i}{2} &lt; b-P_i$. Therefore, $P_i$ is bounded from above by $b$.  Additionally, it is trivial to show that $P_{i+1} &gt; P_i$.  With this in mind, choose an arbitrary $\epsilon$. We need to show that there exists an $n$ such that $P_n &gt; b-\epsilon$. Assume the contrary, that $P_n$ is bounded by $\epsilon$.  Using a similar argument to what which was used before, assume $P_n$ is bounded by $\epsilon$ and then there exists a $P_i$ such that $P_{i+1}\not &gt; P_i$. But this results in a contradiction with (4)  Therefore, $P_n$ is not bounded by $\epsilon$ and we have proved what we want to prove.
"wait on" or "wait for"? Officials are waiting on autopsy results for Oscar-winning actor Philip Seymour Hoffman. New York police suspect a drug overdose, possibly heroin.   I am curious about the preposition on in the sentence, because I usually say "wait for" instead of "wait on". Does it still have the same meaning if I change on to for? Just as follows:     Officials are waiting for autopsy results for Oscar-winning actor Philip Seymour Hoffman. New York police suspect a drug overdose, possibly heroin. There is a book I read in the 1980s or 1990s from a Ca.  library. It was a hard copy not a paperback. It is not "To Outlive Eternity/Tau Zero" by Poul Anderson (I have that one). It may have been written by a woman or someone whose last name starts with an "L".  A large crew of men and women are in a new super fast space ship. They are being chased by someone and they outrun the collapse of the universe, but they have lost their home planet. They watch as the universe is reborn. They go into orbit around a planet that looks like it can support life like theirs. They are moving super fast. They look like a ring of radiation going around the planet. They put down 10 super metal markers with scientific info onto the surface of the planet. They watch the planet develop and form life like the dinosaurs and then evolve into human like life forms.  There are 2 warring countries on the planet. They block or turn off the sun to freeze the planet and force the countries to form one world government. The planet gets freezing cold and the people start to freeze. The countries agree to form one world government. The people on the ship are welcomed to the planet. Writing to NTFS drive in Linux I've got an external USB drive that is formatted with NTFS mounted on ~/tmp.  I can read it just fine, but cannot write to it, even as root (temp is a folder on the USB drive):  [root@Plugbox temp]# touch test touch: cannot touch `test': Permission denied [root@Plugbox temp]#   However, if I do a cat /proc/mounts, it looks like I do have NTFS read/write support:  [root@Plugbox temp]# grep /dev/sdb1 /proc/mounts /dev/sdb1 /root/tmp ntfs rw,relatime,uid=0,gid=100,umask=02,nls=utf8,errors=continue,mft_zone_multiplier=1 0 0 [root@Plugbox temp]#   Finally, if I go to the folder above temp, it looks like I should have plenty of permissions to write to the folder:  [root@Plugbox tmp]# ls -al total 61500764 ... drwxrwxr-x 1 root users           0 Apr 19  2010 temp ... [root@Plugbox tmp]#   What am I missing? Mirror changes to side of cube across all six sides Relationship between $R^2$ and correlation coefficient A borderline between on-topic questions and questions about library recommendation? I have a consideration about new off-topic close reason:     Questions asking us to recommend a tool, library or favorite off-site   resource are off-topic for Stack Overflow as they tend to attract   opinionated answers and spam. Instead, describe the problem and what   has been done so far to solve it.   For example take that question:   It has a close vote with the reason above. But if it would be formed     How to zoom image in an absolute position div to the side?   The answer would be probably also a jQuery plugin or javascript library, but it would follow the patter of most questions about jQuery.  Take that question:   This is the good question about real-life problem. The answer is to use a maven plugin. It is correct, there's no need to reinvent the wheel or rediscover the America.   But  I'm afraid the off-topic text could be misleading and could be misused to close such a questions as that about maven simply because a tool, library or plugin is the answer.   Is it a new guideline that the questions such as cited above should be banned, no matter how they are formed (this would make most how-to-do-in-maven/jquery) questions illegal? Or the close reason should be changed to make more clear what the guidelines are (it's not clear to me now)? Two git repositories in one directory? Is it possible to have 2 git repositories in one directory?  I'd think not, but thought I'd ask.  Basically, I'd like to check in my home directory config files (e.g. .emacs) which should be common across all of the machines I work on, but have a second repository for local files (e.g. .emacs.local), which contains machine-specific configurations.  The only way I can think of to do that is to have the local config in a subdirectory and ignore that subdirectory from the main git repository.  Any other ideas? What award did Leia give Chewie, Han and Luke? What award/medal is this -  I have been searching  to try and find what medal this is, but i have not found it, does anyone know what this medal/award was and why it would be given (eg- bravery, serving in the battle of Yavin)? Linear regression not fitting well I do a linear regression using R lm function:  x = log(errors) plot(x,y) lm.result = lm(formula = y ~ x) abline(lm.result, col="blue") # showing the "fit" in blue     but it does not fit well. Unfortunately I can't make sense of the manual.  Can someone point me in the right direction to fit this better?  By fitting I mean I want to minimize the Root Mean Squared Error (RMSE).    Edit: I have posted a related question (it's the same problem) here:   and the raw data here:    except that on that  x is what is called errors on the present page here, and there are less samples (1000 vs 3000 in the present page plot). I wanted to make things simpler in the other question. Why are there scenes that will not render on GPU but will render fine on CPU? I've already tried GPU Rendering and it seem to be twice as fast as the CPU. However I have some situation where tried to render on the GPU, but it doesn't do anything. When I switch to CPU it renders.  I have some fur texture/materials using particle systems. I wonder if that causes the GPU not to render, I'm not really sure. How to completely disable auto-save and versions in Mac OS X Lion? I hate the automatic saving and versioning in Mac OS X Lion.  Is there a way to completely turn it off?  (This has been asked before , but the answer and the linked article are completely wrong, so I thought I'd try again.) Borel-Cantelli-related exercise: Show that $\sum_{n=1}^{\infty} p_n < 1 \implies \prod_{n=1}^{\infty} (1-p_n) \geq 1- S$. This is supposed to be related to the  (my justification for the independence tag). In Williams' Probability with Martingales, 2BCL is proven and then the following is given as an exercise:  Prove $S \doteq \sum_{n=1}^\infty p_n &lt; \infty \implies \prod_{n=1}^\infty (1-p_n) &gt; 0$ assuming $0 \leq p_n &lt; 1$. Hint: Show that $S &lt; 1 \implies \prod_{n=1}^\infty (1-p_n) \geq 1 - S$.   Proving the hint: I tried to find $\{a_n\}$ s.t. $$\prod_{n=1}^\infty (1-p_n) \geq \prod_{n=1}^\infty e^{a_n} = e^{\sum_{n=1}^\infty a_n} \geq 1 - \sum_{n=1}^\infty a_n \geq 1 - S &gt; 0$$ That is, find $\{a_n\}$ s.t.: 1a. $1 - p_n \geq e^{a_n}$ 1b. $a_n \leq p_n$ I thought of $a_n = \ln(1-p_n)$ (a reason why $p_n &lt; 1$, I guess) Is that right? Edit: Actually, assuming my proof is right, is $S &lt; 1$ used in the proof? If so, where?  From Williams book: Remote desktop without a password I have an HTPC with a user that has no password and autologin.  I want to be able to connect to it via Remote Desktop without a password, so I can login with the original user. (I don't want to add another user just for the mstsc)  Is there a way of doing that?  The OS is Windows 7, 32bit. Extend or remove Data Explorer timeout I know that we don't have automatic login for the Data Explorer, but what's with the automatic timeout/logout?  There have been times that I am logged into the site, maybe I walk away for a little bit, and I come back to being logged out.    Is there a reason for logging us out of Data Explorer?  If so, is it possible to either remove the timeout or extend the period of time before timeout? Sets with no asymptotical density over $\mathbb N$ Let's consider the  defined by:      Take $ A\subset \mathbb N$; define the sequence $x_n= \dfrac{|A\cap[1,n]|}{n}$, and then if $\lim\limits_{n\to\infty} x_n$ exists, call it $d(A)$, the density of $A$ over $\mathbb N$.   My teacher said that the collection of sets that have a natural density is not an algebra, but I don't know why. I can't even find a set that does not have a natural density.  Please help me )= Capacitor Voltage Rating How to find the voltage rating of a capacitor after knowing the capacitance value.  Example, For a power supply, If the input voltage is 60V and output is 5V. The input capacitor(C1) value is 100uf and output capacitor value is 1000uf.  What is the voltage rating for both the capacitors and how to find them. Is smell a bad way of determining whether meat is still good? I have always relied on my sense of smell to determine whether meat (most often chicken) is bad and I thought it was reliable. However, I have just heard that [1] smell is not always reliable, since some of the toxins produced by bacteria won't cause any smell whatsoever and you won't be able to notice that the meat is off until after you've eaten it.   Is this true? If yes, is this true for all meat or only for some? Is there a more reliable way of determining whether meat is still good.  [1] "Heard that" isn't very reliable; so that's why I'm asking the question here. :) Getting Started with Unit Testing Unit testing is, roughly speaking, testing bits of your code in isolation with test code. The immediate advantages that come to mind are:  Running the tests becomes automate-able and repeatable You can test at a much more granular level than point-and-click testing via a GUI    My question is, what are the current &quot;best practices&quot; in terms of tools as well as when and where to use unit testing as part of your daily coding? Lets try to be somewhat language agnostic and cover all the bases. Why does my Nikon D3100 give the error "Lens not attached" with a Vivitar 200 telephoto lens? What is the range of the validity of Coulomb's law? What is the smallest and biggest distance in which  is valid?  Please provide a reference to a scientific journal or book. Just saying that this law is valid from this range to that range is not enough. Which experiments did result to these ranges?