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Topic: ran(EF) contains ~EF(n)
Replies: 9   Last Post: May 1, 2013 10:16 AM

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 William Elliot Posts: 2,637 Registered: 1/8/12
Re: ran(EF) contains ~EF(n)
Posted: May 1, 2013 12:22 AM

On Tue, 30 Apr 2013, Ross A. Finlayson wrote:
> On Apr 30, 8:34 pm, William Elliot <ma...@panix.com> wrote:

> "EF_d(n) is n/d for 0 <= n/d <= 1 for natural integers n, d, as d goes
> to infinity."
>

EF, or f for short, isn't a function, it's a group of functions f_d
with the property, f_d(n) = n/d provided n <= d. Thus to make some
sense of you hand waving:

For d in N, f_d is a function from { 1,2,.. d } into [0,1]
that maps n to n/d.

How are you defining f = lim(d->oo) f_d. Pointwise?
Then f(n) = lim(d->oo) n/d = 0, except that f_d isn't defined
for all n in N.

So you'e defining f_d to be over N mapping n to min{ n/d, 1 }.
Thus f(n) = lim(d-oo) min{ n/d, 1} = 0 for all n in N.