Re: Zero Content


"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

I modified it a little - is it still wrong?

Let I_1, ..., I_(k-1) be intervals such that I_1 = [x_1,x_1+epsilon/k], I_2
= [x_2, x_2+epsilon/k]...
I_(k-1) = [x_(k-1), x_(k-1)+epsilon/k]. Let I_k = [max (x_k, x_(k+1),...),
min(x_k, x_(k+1),...)] Since {x_k} is a convergent sequence then there exist
an epsilon s.t. length of I_k <
epsilon_2. Let epsilon_2 = epsilon/k. 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. The sum of those
lengths will be epsilon.

Tadeusz


.

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