Re: abundance of irrationals!)

"r.e.s." <r.s@xxxxxxxxxxxxxxxx> wrote in message news:<lWB7e.6258$lP1.6049@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx>...

> That produces a *string* of infinite length, namely s = 0.111...,
> (with a 1 in the nth position to the right of the radix point for
> every n in N = {1,2,3,...}. Since no limit is to be taken, s is
> merely an infinite string with no numerical value yet defined.
>
>
> > Take Sum b_k * 10^-k without letting k become actually infinite.
> Rather,
> "Take Sum 2^-k without letting k become actually infinite."
>
> How, exactly?
> Please *define* what you mean by that, without taking a limit.
> (You do know that taking a limit with k -> oo does *not* involve
> "letting k become actually infinite", right?)

It does involve oo. Because we have for any finite natural number n:

n
SUM 1/2^k = 1 - 1/2^n < 1
k = 1

independent of the question how many terms contribute to the sum. The
criterion for obtaining the Sum = 1 is not the infinitude of terms to
add but that in the limit 1/2^n = 0 which is impossible for any
natural number.

Bu you can see this fact more clearly by the sequence 1/k which
requires k = oo to assume its limit 0.

Regards, WM
.

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