Zero Content

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?

Tadeusz


.

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