On Mon, 4 Mar 2013, Ross A. Finlayson wrote: > On Mar 3, 7:22 pm, William Elliot <ma...@panix.com> wrote: > > On Sun, 3 Mar 2013, Ross A. Finlayson wrote: > > > Survey: does anybody find that:
> > > . . lim_d->oo lim_n->d n/d = 1 > > > > > It's clear that it does, for all values of d e N. > > > > > Then, as a function f = n/d from N to R[0,1], d e N, n <= d E N, is it > > > not constant monotone increasing? > > > > Is that f(d) = n/d or f(n) = n/d? > > What's deN and dEN? > > > > > If not increasing, how is lim_n->d n/d = 1? > > > > lim(x->d) x/d = 1, because since f(x) = x/d is continuous, > > lim(x->d) x/d = d/d = 1 > > > > For integer variables n, lim(n->d) f(n) is meaningless. > > Give a definition for it. Wouldn't it be the same as f(d)? > > > > > Answer: it is. > > Only if n is a real variable. > > Thanks, that should read d e N, "d in N" or "d element of N", and > f_d(n) = n/d.
d in N is easier read and prefered.
> Then, it's clear that as d->oo from 1, the range is 0, 1/2, and 1, > then 0, 1/3, 2/3, and 1, and so on, all the fractions or ratios > (reduced) or rational numbers in [0,1].
The range of what?
> For each d, f is constant monotone increasing, as a non-negative real- > valued step function, for equal differences in magnitude in the domain > seeing equal differences of magnitude in the range, and > correspondingly for a natural-valued function. > Huh? Do you mean f_d is constantly monotone increasing? Well yes, it's linear.
> Here, f is increasing, for each d. As d increases, the constant > monotone difference: decreases. In the limit, delta, for the > difference, goes to zero, d for denominator, also for: differential.
Indeed, as df_d(x)/dx = 1/d, lim(d->oo) df_d(x)/dx = 0.
> In the limit, 1/d goes to zero, but the function is still increasing, > as the limit of f exists and is greater than zero. (Let n range in > the reals and apply the intermediate value theorem, f is increasing > else it wouldn't have a limit greater than zero.) > > Does n/d have these simple analytical properties? > Yes, that's a verbose characteristic of f_d which I you're talking about and not some arbitrary function f.