Re: number with Pi.

In article <ep53u3$at$1@xxxxxxxxxxxxxxxx>,
"mina_world" <mina_world@xxxxxxxxxxx> wrote:

Hello sir~

Find the distance between {(2n + (1/2))*Pi | n in N} and N.

This is very much like the following: If a > 0 is irrational,
then d({na : n in N}, N) = 0. To see this, assume first a > 1.
Let [] denote the greatest integer function. Because a is
irrational, the points na - [na], n in N, are all distinct, hence
comprise an infinite subset E of (0,1). Let eps > 0. Because E is
infinite, there are distinct points in E < eps in distance apart.
Ie, there exist m < n in N such that |(na - [na]) - (ma - [ma])|
= |(n-m)a - ([na]-[ma])| < eps. Because a > 1, [na]-[ma] is in
N, so we're done in this case. If a < 1, choose n_0 in N such
that n_0*a > 1. The above shows d({n(n_0*a) : n in N}, N) = 0, so
we're finished.
.