|
|
Re: Cauchy sequence
Posted:
Dec 5, 2003 1:09 AM
|
|
On Thu, 04 Dec 2003 16:48:06 +0000, Charlie Johnson wrote:
> Hi all,
>
> All the Analysis books I own only discuss Cauchy sequences in the Abstract.
> But, I can't seem to figure out exactly how I am to apply them. For
> example,
>
> How do I show that the sequence
> {sqrt(2), sqrt(2*sqrt(2)), sqrt(2*sqrt(2*sqrt(2))), ....} is Cauchy?
>
> How do I show that for any n,m > N |a_n - a_m| < epsilon? For any number
> two elements in the sequence above, |a_n - a_m| > epsilon. (???)
The Cauchy criterion isn't too useful concretely because determining the N
above may be difficult -- did you see the longish thread a couple of weeks
ago about very large numbers? A number of sequences came up that grow
really rapidly. Now suppose that we have any increasing sequence S of
positive integers, with s_1 = 1. A fast-growing s lets us find a slow-
growing S by taking S_n to be the largest integer k for which s_k <= n.
So what? Well, n-> a_S_n is Cauchy if n -> a_n is Cauchy (and has the
same limit). So N as a function of epsilon may grow VERY rapidly as
epsilon -> 0.
For a really pathological example, let a_n = 1/n, and let bb_n be the
n-th busy-beaver number. Now take BB_n to be the largest integer k for
which bb_k <= n. Now the sequence a_BB_n converges to 0, and so it's
Cauchy ... but it is not _possible_ to compute N given epsilon (since
such a computation would allow us to bound, and so to compute, the
sequence bb).
|
|
