Let {$\displaystyle x_{k}$} be a converging sequence in R. Show that {$\displaystyle x_{1}$, $\displaystyle x_{2}$...} has zero content.
I'm not even sure what the term zero content means?
Thanks
Content is a generalization of measure: http://en.wikipedia.org/wiki/Content_(measure_theory)
You must be working with a particular content here, though, since the function that assigns 1 to a subset of R that contains 0, and 0 to every other subset is a content, and under this content any sequence that contains 0 has content 1.
You have to prove that for every $\displaystyle \epsilon>0$ there exists a finite family of intervals $\displaystyle \{(a_i,b_i):i=1,\ldots,n\}$ which depends on $\displaystyle \epsilon$ such that:
(i) $\displaystyle \{x_1,x_2,\ldots\}\subset \bigcup_{i=1}^{n}{(a_i,b_i) }$
(ii) $\displaystyle \sum_{i=1}^{n}(b_i-a_i)<\epsilon$
Hint :
As the sequence has a finite limit $\displaystyle l$, for every $\displaystyle \epsilon>0$ only a finite numbers of $\displaystyle x_i$ do not belong to $\displaystyle (l-\epsilon,l+\epsilon)$
Fernando Revilla