Re: Zero Content
- From: David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
- Date: Mon, 26 Nov 2007 06:11:34 -0600
On Sun, 25 Nov 2007 12:27:06 -0500, "Tadeusz Jordan"
<tedjj@xxxxxxxxxxxxxx> wrote:
"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.
The word "any" should not be used in definitions like this,
because "for any epsilon > 0 ..." could be read as meaning
"for every epsilon > 0..." or "there exists epsilon > 0 such
that...".
In fact Z has zero content if for every epsilon > 0 there
is a finite collection of intervals such that [etc].
Hence you proof is wrong - at one point you say
"there exists epsilon > 0 such that..." - you need to
prove that _every_ epsilon > 0 has a certain property,
not just that there exists an epsilon > 0 with this property.
Could you tell me where did I make a mistake in my attempt to prove it?
************************
David C. Ullrich
.
- References:
- Zero Content
- From: Tadeusz Jordan
- Re: Zero Content
- From: David C . Ullrich
- Re: Zero Content
- From: Tadeusz Jordan
- Zero Content
- Prev by Date: Re: The infintely small number b
- Next by Date: Re: Zero Content
- Previous by thread: Re: Zero Content
- Next by thread: Re: Zero Content
- Index(es):
Relevant Pages
|
