The sequence looks Cauchy. But I do not know how to approach the proof.
$|a_{n+1}-a_n| \leq \frac{1}{2} |a_{n}-a_{n-1}|$. I'm trying to show that $(a_n)$ is convergent by showing that $(a_n)$ is Cauchy.  let $|a_2 - a_1 |  =x$.   $|a_3-a_2| \leq \frac{1}{2}x$  $|a_4-a_3| \leq \frac{1}{2}|a_3-a_2|\leq \frac{1}{4}x$  In general, for every $n$:  $|a_{n+1}-a_n| \le \frac {1}{2^{n-1}}x$  Let $\epsilon &gt;0$  for every $m,n &gt; 0 $, say $m&gt;n$ the following holds:  $$|a_m-a_n| = |a_m-a_{m-1} + a_{m-1} - a_{m-2} \dots a_{n+1}- a_n| \le$$  $$|a_m-a_{m-1}| + |a_{m-1} - a_{m-2}| + \dots |a_{n+1}-a_n| \leq \frac{x}{2^{m-2}} + \frac{x}{2^{m-3}} + \dots + \frac{x }{2^{n-1}}$$.  I can say that for big enough $n,m$, $|a_m-a_n|&lt;\epsilon$ and therefore, this sequence is Cauchy? It sounds not good enough to me yet I don't know what else to do here.
Restrict the scope of a command to the execution of another command Numbers to the Power of Zero Who uses Dual_EC_DRBG? Recent news articles have suggested that the NSA may be involved in trying to influence the cryptography in public standards or commercially deployed software, to enable the NSA to decrypt the encrypted traffic.  For example, .  The New York Times article specifically mentions a 2006 NIST standard, where two Microsoft engineers later discovered a serious problem in the scheme.  The New York Times article explains how the standard was written by the NSA, who pushed the scheme hard:     Simultaneously, the N.S.A. has been deliberately weakening the international encryption standards adopted by developers. One goal in the agency’s 2013 budget request was to “influence policies, standards and specifications for commercial public key technologies,” the most common encryption method.      Cryptographers have long suspected that the agency planted vulnerabilities in a standard adopted in 2006 by the National Institute of Standards and Technology, the United States’ encryption standards body, and later by the International Organization for Standardization, which has 163 countries as members.      Classified N.S.A. memos appear to confirm that the fatal weakness, discovered by two Microsoft cryptographers in 2007, was engineered by the agency. The N.S.A. wrote the standard and aggressively pushed it on the international group, privately calling the effort “a challenge in finesse.”    After some sleuthing, I'm pretty certain this is a reference to the  pseudorandom number generator scheme described in NIST SP 800-90.  The weakness is that Dual_EC_DRBG appears to contain a backdoor, and anyone who knows the backdoor can totally break the PRNG.  The weakness was first described in  and was subsequently .  As a consequence, there are now good reasons to suspect that Dual_EC_DRBG contains a NSA backdoor, and the NSA might be able to spy on anyone who is using Dual_EC_DRBG.  My questions:   Who uses Dual_EC_DRBG? Do any commercial products use Dual_EC_DRBG? Does anyone use it? Dark matter and dark energy references I've been looking for questions about dark matter, and I've read some very interesting answers. However, I desire too look into it deeply.   This is not actually a question. I'm asking the community to recommend interesting references to understanding dark matter and dark energy.   I accept all sort of references: notes, books, scientific papers etc.  Let us assume some background on classical physics, thermodynamics and basics about quantum theory. Comparison of backup tools This question exists because it has historical significance, but it is not considered a good, on-topic question for this site, so please do not use it as evidence that you can ask similar questions here.  While you are encouraged to help maintain its answers, please understand that "big list" questions are not generally allowed on Ask Ubuntu and will be closed per the .   Backup is incredibly important. Obviously there's no best backup tool, but a comparison of the options would be very interesting.   Graphical Interface? Command line? Incremental backups? Automatic backups? Install method: In standard repositories? PPA? If no exact time (such as 15:00 or 21:00) is specified about the deadline and only the date itself is given, does that mean that the Deadline date is inclusive or exclusive?  For example, if the deadline for my Research Project is 10th October (no other exact times are specified) does that mean that I can turn it in on 10th October before it turns into 11th October? Or should I submit it on 9th October before it turns into 10th? How do you repair an "input/output error" in an NTFS partition? I am an Indian citizen. I am flying from Toronto to New Delhi. I have two transits: one at Lisbon airport and the second one at Madrid airport.  While the Consulate of Portugal has assured me that I do not need an airport transit visa during my 2 hours stay in Lisbon airport, I did not get a definitive answer from the consulate of Spain in Toronto.   At Madrid airport, I need to make a transfer from Terminal 2 to Terminal 1.   Does anyone here know whether I need to pass through immigration while making this transition in Madrid and if so, do I need a Schengen visa for that? How do idle games handle such large numbers? Just wondering how games such as Tap titans and Cookie Clicker handle such large numbers.  I am trying to implement an idle game however the largest number format supported by C# is decimal.   I am looking to support up to 10^500; that needs to be floating point  How could I handle this?  P.S. it needs to be cross platform i.e. pc, mac, ios, android and compatible with Unity What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by  $$x_1=r \cos  \theta$$ $$x_2=r \sin  \theta$$  For $n=3$, the analogue would be the ordinary spherical coordinates $r,\theta ,\varphi$, related to the Cartesian coordinates $x_1,x_2,x_3$ by  $$x_1=r \sin  \theta  \cos  \varphi$$ $$x_2=r \sin  \theta  \sin  \varphi$$ $$x_3=r \cos  \theta$$  So these are my questions: Is there an analogue, or several, to spherical coordinates in $n$-dimensions for $n&gt;3$? If there are such analogues, what are they and how are they related to the Cartesian coordinates? Thanks. What are logic gates? So from what I understand transistors are one of the most important electronic elements for computers because of TTL (Transistors-Transistors Logic).  Is this true? 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Finding the dual basis for the space of polynomials of degree $\leq$ 2 Let $V$ be the space of polynomials of degree at most $2$, with basis $\{1,x,x^2 \}$.Find the dual basis to $\{1,x,x^2\}$.   I know that the dual basis of $\{f_1,f_2,f_3\}$ is the basis $\{f_1^*,f_2^*,f_3^*\}$ where $f_i^*(a_1f_1,\ldots,a_3f_3)$ but I'm not sure how to actually compute the dual basis. Replaced wrong car in garage in GTA Online Remove the Yoast SEO Post Metabox Yoasts SEO plugin adds a metabox to the post edit screen. I'm trying to remove this for users who aren't editors or above.  I've tried putting a remove_meta_box call in on admin_init, trying to remove the action on $wpseo_metabox but to no avail.  How do I remove this metabox without requiring user intervention (the user should never know the metabox existed, so clicking on screen options is not an option ) How and why do you use an image histogram? 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