# Thread: zero content?

1. ## zero content?

Let { $x_{k}$} be a converging sequence in R. Show that { $x_{1}$, $x_{2}$...} has zero content.
I'm not even sure what the term zero content means?
Thanks

2. 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.

3. Originally Posted by calculuskid1
I'm not even sure what the term zero content means?

You have to prove that for every $\epsilon>0$ there exists a finite family of intervals $\{(a_i,b_i):i=1,\ldots,n\}$ which depends on $\epsilon$ such that:

(i) $\{x_1,x_2,\ldots\}\subset \bigcup_{i=1}^{n}{(a_i,b_i) }$

(ii) $\sum_{i=1}^{n}(b_i-a_i)<\epsilon$

Hint :

As the sequence has a finite limit $l$, for every $\epsilon>0$ only a finite numbers of $x_i$ do not belong to $(l-\epsilon,l+\epsilon)$

Fernando Revilla