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Topic: Discussion with WM - Frustration reaches boiling point (What is not clear?)
Replies: 198   Last Post: Jul 10, 2014 12:04 AM

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 William Elliot Posts: 1,494 Registered: 1/8/12
Re: Discussion with WM - Frustration reaches boiling point (What is
not clear?)

Posted: Jul 1, 2014 11:15 PM

> I have been discussing with WM about the following sequence of sets
> s_n = {-1/n, 1/n}.
> The standard product topology tells us that
> lim s_n = { }

What product?

> My "conviction" that this is intuitive follows from these two premises:
> 1. r->0 (0,r) -> { }
> 2. Limits should pass through subset relations.
> Under premise 1 and 2, I have the following (note the usage of <= for subset)
> {} <= {-1/n, 1/n} <= (-2/n, 0) u (0,2/n) <= (-r,0) u (0,r) , n = floor[2/r] > + 1
> {} <= lim {-1/n, 1/n} <= lim (-2/n, 0) u (0,2/n) <= lim (-r,0) u (0,r) = {}
> Here, the limit is taken as r->0
> So it follows that
> lim r->0 {-1/n(r), 1/n(r)} ={ }, but n(r) -> oo as r->0
> My original sequence s_n = {-1/n,1/n} appears as a subsequence as r->0 in
> the above limit. (There is hidden third premise: subsequence of a convergent
> sequence converges.)
> Therefore,
> lim n->oo {-1/n, 1/n} = { }

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