Re: Dedekind Cuts, Fundamental Sequences: why?

In article <LN6dnWbFja4Qg_XbnZ2dnUVZ_vmqnZ2d@xxxxxxxxxxxxx>,
Hatto von Aquitanien <abbot@xxxxxxxxxxxxxx> wrote:


And no, neither "convergent" nor "Cauchy" requires that
abs(x_n - x_m) be monotone decreasing.

(In fact it's not at all clear what it _means_ to say that
this is monotone decreasing, since there are two indices.
But in any case no, there's nothing about monotonicity in
the definitions of "Cauchy" or "convergent".)

But there is a theorem which says it will be confined to a monotone
decreasing function.

WHAT will be "confined to a monotone decreasing function"?

There is nothing in the definition of Cauchy sequences which requires
any sort of monotonicity.

Do you mean that for a Cauchy sequence, x_n, abs(x_n - x_m) converges
to zero as min(m,n) increases?

If so, you have expressed yourself misleadingly.
.