Re: Zero Content
- From: "Tadeusz Jordan" <tedjj@xxxxxxxxxxxxxx>
- Date: Sun, 25 Nov 2007 12:27:06 -0500
"David C. Ullrich" <ullrich@xxxxxxxxxxxxxxxx> wrote in message
news:pi5jk39a9dii3ni3cfj7jm2g1cmb64dqo7@xxxxxxxxxx
On Sat, 24 Nov 2007 19:08:04 -0500, "Tadeusz Jordan"
<tedjj@xxxxxxxxxxxxxx> wrote:
Hello,
I am studying for my exam and I have some questions. Could anyone help me?
Let {x_k} be a convergent sequence in R. Show that the set {x_1, x_1,...}
has zero content.
When I attempted to solve this problem I obtained this:
Let I_1, ..., I_(k-1) be intervals such that I_1 = {x_1}, I_2 = {x_2}...
I_(k-1) = {x_(k-1)}. Let I_k = {x_k, x_(k+1),...} Since {x_k} is a
convergent sequence then there exist an epsilon s.t. length of I_k <
epsilon. k is a finite number so there are finitely many intervals and
{x_1,
x_2,...} is a subset of the union of those intervals.
Is this proof correct?
No. What's the definition of "zero content"?
In particular, does the definition read
"S has zero content if there exists an epsilon > 0
such that..."?
Tadeusz
************************
David C. Ullrich
The definition of zero content from my textbook is:
A set Z in R is said to have zero content if for any epsilon > 0 there is a
finite collection of intervals I_1, ..., I_L, such that
i) Z is in the union of those intervals, and
ii) the sum of the lengths of those intervals is less than epsilon.
Could you tell me where did I make a mistake in my attempt to prove it?
.
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