For two numbers $a$ and $b$ with $0 \leq a\leq b $ we have that: $a_{n+1} =\sqrt{a_nb_n} \quad$ , $\qquad b_{n+1} = \frac {a_n+b_n}2 \quad,$ $\qquad a_0 = a, \quad b_0=b $ And I have to prove for $n\ge1$ that $b_n-a_n &lt; (\frac {1}2)^n(b-a) \quad $ if $a\neq b$ In previous exercises I've proved that: (i) $\sqrt{ab} \leq \frac{a+b}2$ $\quad$ for $\; 0\leq a \leq b \;$ with equality if and only if $a=b$ (ii) $ a &lt;\sqrt{ab} \; $ and $\frac{a+b}2 &lt;b \; $ for $0 \leq a&lt;b$ By using (i) and (ii) I have done this so far: $b_n-a_n =\frac{a_{n-1}+b_{n-1}}2 - \sqrt{a_{n-1}b_{n-1}} $ $\qquad \quad &lt; \frac{a_{n-1}+b_{n-1}}2-a_{n-1}=\frac{1}2(b_{n-1}-a_{n-1}) \quad$ for $n\ge1$ I'm stuck from here. (Edited)
I am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all $n\in\Bbb{N}$    To prove is that both sequences converge and that they have the same limit. I don't know how to show this. I have spent 2 hours on this, no sign of success
When performing apt-get update, I get the following error:   root@ADS3-Debian6:/home/aluno# apt-get update Atingido http://sft.if.usp.br squeeze Release.gpg Ign http://sft.if.usp.br/debian/ squeeze/contrib Translation-en Ign http://sft.if.usp.br/debian/ squeeze/contrib Translation-pt Ign http://sft.if.usp.br/debian/ squeeze/contrib Translation-pt_BR   (...)  Obter:10 http://security.debian.org squeeze/updates/non-free i386 Packages [14 B] Baixados 612 kB em 4s (125 kB/s)                     Lendo listas de pacotes... Pronto There is no public key available for the following key IDs: 8B48AD6246925553 Which should I use on my DB? - Clustered Index or Non Clustered Index or Both I just need advice on how I am going to optimize my database.  Date ---- Time ---- Area ---- Block ---- Data1 ---- Data2 ---- Data3 ---- DataN   I have that format of tables on my database. Each tables has 30-days of records with hundred thousands of records each data. Uploading of data to database is every morning. The data of yesterday will be uploaded.  The most common queries to run is grouped by Date or Date and Area or Date and Area and Block. Now, in order to make queries faster, I found out using indexes. I used indexes before. But that was when I am using a primary key. In this table, I didn't use primary key because it is totally unnecessary.   Now, I am really confused on how I am going to optimize this. Based on  on Column Considerations part, clustered index is not a good choice on Columns that undergo frequent changes. I am really confused.  Please help me if I need to use clustered index here or just non clustered or both. As a new user on a site, what's the best way to ensure your question will be positively received? $\DeclareMathOperator{\tr}{tr}$Let's consider a $n \times n$ matrix and the sequence of traces $\tr(A)=\tr(A^{2})= \ldots = \tr(A^{n})=0$. How to prove that $A$ is a nilpotent matrix (a matrix so that $A^{k} \times u = 0$ for all $u \in V$ and for some $k$)?  Maybe it would be reasonable to consider $A$'s Jordan form (we assume that the action occurs over an algebraically closed field, such as $\mathbb{C}$). For example, suppose that we have found it and $A'$ consists of $m$ blocks of sizes $r_{1}, \ldots r_{m}$ ($r_{1}+r_{2}+\ldots+r_{m}=m, r_{i} \in \mathbb{Z}_{+}$) with $\alpha_{1}, \ldots \alpha_{m}$ as corresponding eigenvalues to each of the blocks. Then, according to the problem will get the system  $$\begin{align} r_{1}\alpha_{1}+&amp;\ldots+r_{m}\alpha_{m}=0 \\ r_{1}{\alpha_{1}}^{2}+&amp;\ldots+r_{m} {\alpha_{m}} ^{2}=0\\ &amp;\ ~\,\vdots \\ r_{1}{\alpha_{1}}^{n}+&amp;\ldots +r_{m}{\alpha_{m}}^{n}=0  \end{align}$$  But the system's analysis over $\mathbb{C}$ seems to be too much complicated.  Actually, the problem must have more or less simple solution.  Any piece of advice would be much appreciated. Today I was informed by VFS that my tourist visa for UAE got approved. I'm travelling to UAE on 16th December 2014 and returning on 7th Jan 2015. But when I checked the "valid until" date on the visa it states 30 December 2014.    Does this date mean I should return before December 30th? Should I get it corrected from VFS? Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is it true that $I=(0)$ ?  I know that a surjective ring endomorphism of a Noetherian ring is also injective, and since there is a natural surjection from $R$ onto $R/I$ we get a surjection from $R$ onto $R$, but the problem is I can not determine the map explicitly and I am not sure about the statement. Please help. Thanks in advance. What set of books makes a complete starter set for 4e? I've asked several questions about 4th Edition D&amp;D because I may be able to start a game. However, I'd need to purchase (some) materials. I am aware of the 4 book set (core) and the expected Essentials Red Box.  So now I need to know: presumably buying either the 4 book set or essentials will be enough to get my game on. Would there be something 'lacking' though? Would I be missing character types, for instance, as I've seen that the splatbook strategy is still in effect?  I'm an experienced player and GM and plenty creative, so many holes I'd be able to fill in myself. I just don't want to get sucked into too much financial impact. How to understand the notion of a differential of a function Why is the word 'number' abbreviated to 'No.' in UK English and '#' in American English? Why the disparity? And why use 'No.'? Is it from the French?  And the hash or pound sign seems a weird choice too, is there a history or any reason involved? Serializing anonymous delegates in C# I am trying to determine what issues could be caused by using the following serialization surrogate to enable serialization of anonymous functions/delegate/lambdas.   // see http://msdn.microsoft.com/msdnmag/issues/02/09/net/#S3 class NonSerializableSurrogate : ISerializationSurrogate {     public void GetObjectData(object obj, SerializationInfo info, StreamingContext context)     {         foreach (FieldInfo f in obj.GetType().GetFields(BindingFlags.Instance | BindingFlags.Public | BindingFlags.NonPublic))             info.AddValue(f.Name, f.GetValue(obj));     }      public object SetObjectData(object obj, SerializationInfo info, StreamingContext context,                                 ISurrogateSelector selector)     {         foreach (FieldInfo f in obj.GetType().GetFields(BindingFlags.Instance | BindingFlags.Public | BindingFlags.NonPublic))             f.SetValue(obj, info.GetValue(f.Name, f.FieldType));         return obj;     } }     Listing 1 adapted from   The main issue I can think of that might be a problem is that the anonymous class is an internal compiler detail and it's structure is not guaranteed to remain constant between revisions to the .NET Framework. I'm fairly certain this is the case based on my research into the similar problem with iterators.  Background  I am investigating the serialization of anonymous functions. I was expecting this not to work, but found it did for some cases. As long as the lambda did *not&amp; force the compiler to generate an anonymous class everything works fine.   A SerializationException is thrown if the compiler requires a generated class to implement the anonymous function. This is because the compiler generated class is not marked as serializable.  Example  namespace Example {     [Serializable]     class Other     {         public int Value;     }      [Serializable]     class Program     {         static void Main(string[] args)         {             MemoryStream m = new MemoryStream();             BinaryFormatter f = new BinaryFormatter();              // Example 1             Func&lt;int&gt; succeeds = () =&gt; 5;             f.Serialize(m, succeeds);              // Example 2             Other o = new Other();             Func&lt;int&gt; fails = () =&gt; o.Value;             f.Serialize(m, fails); // throws SerializationException - Type 'Example.Program+&lt;&gt;c__DisplayClass3' in Assembly 'Example, Version=1.0.0.0, Culture=neutral, PublicKeyToken=null' is not marked as serializable.         }     }   Listing 2  This is similar to the issue of trying to serialize iterators and I had found the following code in a previous search (see ) Using the code from Listing 1 and an ISurrogateSelector I was able to successfully serialize and deserialize the second failing example.  Objective  I have a system that is exposed via a web service. The system has a complex but small state (many objects, not a lot of properties per object). The state is saved in the ASP.NET Cache, but is also serialized to a BLOB in SQL in case of cache expiration. Some objects need to execute arbitrary "events" upon reaching some condition. Hence they have properties accepting Action/Func objects. Contrived example:      class Command     {         public Command(Action action, Func&lt;bool&gt; condition);     }   Somewhere else      void DoSomethingWithThing(Thing thing)     {         state = Store.GetCurrentState();          Command cmd = new Command(() =&gt; thing.Foo(), () =&gt; thing.IsReady())         state.Add(cmd);          Store.Save(state);     } Is there any UNIX variant on which a child process dies with its parent? How to move a window if titlebar is overlapped by Ubuntu menu Somehow the title-bar of chrome was overlapped by Ubuntu menu-bar like attached image.  In this case how to move this chrome window? Measuring accuracy of latitude and longitude? How to manually mount a partition? There are many questions on automatically mounting or problems encountered during mounting that I've read here. But none on how to manually mount a partition. How to upload an mp3 to Facebook? I know it's possible to upload videos to Facebook. Is it possible to upload an mp3 to Facebook using their user interface? Looking at the kids which - or that/who? We look at the kids which/that/who skillfully use tablets with the help of their small fingers and we understand that to surprise the younger generation will be more difficult each year.   Is the relative pronoun which acceptable here, or does it look totally out of place, because it befits a "non-restrictive relative clause" more?   It seems to me that the pronoun that looks best in the bolded position, with who an acceptable second choice. But it's hard to pin down why. In the  library, they are overloading the () operator.  Is this a convention in C++? For callbacks, etc.?  I have seen this in code of a co-worker (who happens to be a big Boost fan). Of all the Boost goodness out there, this has only led to confusion for me.  Any insight as to the reason for this overload? How do I make a 2D game resolution independent? Non-existence of a bijective analytic function between annulus and punctured disk when to used had been and have been My friend just answered to one of my important questions. Now how should I reply to him?        That question had been in my head for a long time, finally, I got my answer      or        That question has been in my head for a long time, finally, I got an answer.      Basically, I am confused with had and have. Since that question is not anymore in my head after having got an answer, what should be used? (I am also confused whether it is possible to use having instead.)