# Calculus- problem with subsequences

Let $\{a_n\}$ be a sequence and $k$ a natural number so that: $\{a_{nk}\}$ , $\{a_{nk+1}\},\ldots, \{a_{nk+(k-1)}\}$ - every sub-sequence converges to the same limit $L$.

I need to show that $\{a_n\}$ itself converges to $L$.

It is pretty obvious to me why, regarding the fact that from $N=k$, every $a_n$ belongs to one of the sub-sequences above, so we have $N$ that from it and on all numbers will close to $L$, close as we want.

I just don't remember how make it to a formal prove.

I tried to assume the opposite and get Contradiction, but It didn't work.

Thank you

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Hi -- you've asked quite a number of questions now. I think it's about time you familiarize yourself with how to format them so people don't have to keep wading through this mangled notation to read them. This might help you to get started: meta.math.stackexchange.com/questions/1773/… – joriki Apr 4 '11 at 17:44
@joriki: i'm trying to paste what I've wrotr in codecogs.com/latex/eqneditor.php but it doesn't work correctly, what seems to be the problem? is there any special way to paste it? – user6163 Apr 4 '11 at 18:08
@Nir: You can look at what other people do to format by right-clicking on a formula and selecting "View Source". In any case, you should know that it is extremely bad practice to use capital and lower case letters interchangeable; most mathematicians would not consider K and k to necessarily represent the same variable, nor n and N, yet you use them interchangeably (and do this often). – Arturo Magidin Apr 4 '11 at 18:10
@Nir: You need to enclose any $\TeX$ you enter in dollar signs, like this: \$x\$. More generally, as suggested in the thread I linked to above, look at other questions (and answers) by clicking on "edit" (you don't have to be the author; you don't even have to be allowed to edit them; you can always get the source by clicking on "edit"). – joriki Apr 4 '11 at 18:11