Re: Question about closed and compact sets

From: The World Wide Wade (waderameyxiii_at_comcast.remove13.net)
Date: 09/16/04 

Date: Thu, 16 Sep 2004 14:48:46 -0700

In article <Tnm2d.16329$QJ3.8077@newssvr21.news.prodigy.com>,
 "W. Dale Hall" <mailtowd-hall@pacbell.net> wrote:

> Agapito Martinez wrote:
>
> > Consider 2 sets, both subsets of the set of all rationals Q:
> >
> > A = {p: 2< p^2 <3, p rational}

> A is not compact. Consider the open covering of A by the intervals
>
> I_n = (Ln , Rn) = { q in Q | Ln < q < Rn }
>
> and where:
> Ln is the least number of the form N/10^n
> strictly greater than sqrt(2)
> Rn is the greatest number of the form N/10^n
> strictly less than sqrt(3)
>
> as follows:
> L1 = 1.5, L2 = 1.42, L3 = 1.415, ... (decreasing to sqrt(2))
> R1 = 1.7, R2 = 1.73, R3 = 1.732, ... (increasing to sqrt(3))
>
> That is, A > ... > I_(n+1) > I_n > I_(n-1) > ... > I_1
>
> and A = union_n(I_n).
>
> Notice that this covering has no finite subcover. Thus A cannot be
> compact.

Why not just consider the open cover (-oo, sqrt(3) - 1/n), n = 1, 2, ... ?