Consider a convergent sequence  $a_1,a_2,a_3\cdots a_n$ tending to a limit A. Now consider the sequence $K_1,K_2,K_3 \cdots K_n$ such that $K_n =\cfrac {a_1+a_2+...a_n}n$. Now what I guess is that as an tends to infinity $K_n$ tends to $a_n$.  What I can say is that the sequence $K_n$ is a bounded one. As such it must have a convergent sub sequence. Is this sequence also convergent.  I tried to resolve this by first trying to apply squeeze theorem as $K_n$ always lies between the minimum and maximum of that sequence. But I could not meet up rigor or any clear result.   Next I tried to apply Cauchy's principle, but here I don't think that one can reach to a strong consequence except few suggestive apprehensions.  I hope someone can guide me on this problem with a proof.
Prove that if $\lim_{n \to \infty}z_{n}=A$ then:  $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n})$$ where $N$ is value of $n$ for which $|A-z_{n}|&lt;\epsilon$ then taking the limit of this sum devided by $n$ , and noting that the second sum is as close as you wish to $nA$ while the first is as close as you wish to $0$. Not sure if this helps....
How to measure time taken by a function to execute I need to get execution time in milliseconds.  I originally asked this question back in 2008. The accepted answer then was to use  However, we can all agree now  that using the standard  API is more appropriate. I am therefore changing the accepted answer to this one. Multiple Inheritance in C# Add new alias to Gmail without SMTP (forwarding-only address) I used to be able to go to Accounts and Import → Add another email address you own and add an email account that's only a forwarding address. That is, it's not a mailbox and I don't have SMTP access to it.  Gmail would send me a confirmation email to that address, I'd confirm it's mine, and done. Now it only gives me the option to add SMTP credentials.  Did they stop supporting the previous method? Is there any other way I can get that option back? The sum $\sum_{k=1}^nk(k!)$ can be easily calculated by noting $k(k!)=(k+1)!-k!$. Is there a way to calculate the sum nicely using a combinatorial argument. Is it possible to notice it is $(n+1)!-1$ combinatorially? A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$ "Outside" or "outside of"? Can you please explain this to me?          Could you please explain this to me?   I am unable to figure out which to use which situation. I did google, and some posts say they are both the same, even if the second one is more formal. Other posts say could is the past tense of can, but in the above example, I don't think could is used as past tense of can.  What is correct way to use these words? Intersections of all subgroups is a nontrivial subgroup, so every element has finite order. I need help to prove this result:  "Let $G$ be a group such that the intersection of all its subgroups other than $\{1\}$ is a subgroup different from $\{1\}$. Then all its elements have a finite order".  I know I must think of an element $g$ of infinite order. That will imply that every subgroup has infinite order (because it will contain an element like $g^k$). After this, I don't know which step I can take. Can someone help? Why Dumbledore kept mum until the very end of the "Deathly Hallows"? execute command on a defined range of directories Is it possible to 'partition' a directory listing - say in to blocks of some number, to perform different actions on each 'range'?  For example, lets say I have the following directories inside another folder:  $ ls test abc/ def/ hij/ klm/ nop/ qrs/ tuv/ wxy/ zzz/   And I want to perform some action on the first 3 directories, another on the second 3, and so on.  My thought was that perhaps a loop across numbers would work, based on the output of something like ls | nl (but before anyone mentions it, I know parsing ls is a no-no!)  This obviously doesn't work, but illustrates my point I hope:  for dir in `ls | nl`;  do     do-something-with ${dir{1..3}} # Where $dir has taken on some numerical value linked to the folder)     do-something-with ${dir{4..6}}  # And so on...     do-something-with ${dir{n-3..n}}  done   The folders I intend to actually do this on, can be worked on in any order (i.e. the final splits can be totally arbitrary), but they have no logical naming consistency - by which I mean they can't be organised sensibly alphabetically or numerically based on any key within the directory name themselves. "Oriented" vs. "orientated" A great Python IDE for Ubuntu Possible Duplicate:          I want to develop Python apps in Ubuntu, and I'm tired of working with gedit one time and another.  So I'm trying to find a great Python IDE (a true IDE, not just a syntax-highlighting editor) on the Ubuntu repositories. Which one do you recommend me? Relative performance of std::vector vs. std::list vs. std::slist? 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I was wondering whether the following (weaker) statement holds:  Let $X$ be a Banach space and suppose $A,B,C_0\subset X$ are bounded, closed, convex and non-empty subset. Do we then have $$A+C_0=B+C_0\implies A=B?$$ My group of 8 level 3 players are about to run through the Thundertree Ruins and will probably have a hand at fighting the dragon. My question stems from the fact that a single poison breath attack from the dragon will outright kill all but 1 player assuming they fail their saving throws.  I understand this encounter is supposed to be ridiculously deadly, but I don't want to be punishing my players for testing the lengths at which they can fight stuff. A fight should be challenging and they should be able to try and run away, but instant obliteration seems a bit much.  All PCs are level 3 (HP in parenthesis):   Wood elf moon druid  (27) dwarven frenzy barbarian (44) fire genasi archer ranger (37) (pseudodragon companion (12)) wood elf assassin rogue (27) human wild magic sorcerer (22) dragonborn fighter (29) human monk (24) human fighter (32) Java Round up Any Number Thailand visa exemption: any rule limiting number of entries per year? Effect on Google rankings if my site is down?