§ An example of a sequence whose successive terms get closer together but isn't Cauchy (does not converge)
§ The problem
Provide an example of a sequence
such that ,
but . That is,
proide a series where the distances between successive terms converges to zero,
but where distances between terms that are "farther apart than 1" does
not converge to 0. That is, the sequence is not Cauchy.
§ Regular solution: Harmonic numbers
The usual solution is to take the harmonic numbers,
. Then, we show that:
§ Memorable solution: logarithm
We can much more simply choose . This yields the simple
while on the other hand,
I find this far cleaner conceptually, since it's "obvious" to everyone
that diverges, while the corresponding fact for
is hardly convincing. We also get straightforward equalities everywhere,
instead of inequalities.
I still feel that I don't grok what precisely fails here, in that, my intuition
still feels that the local condition ought to imply the Cauchy condition:
if tells to not be too far, and tells ,
surely this must be transitive?
I have taught my instincts to not trust my instincts on analysis, which is a
shitty solution :) I hope to internalize this someday.
EDIT: I feel I now understand what's precisely happening
after ruminating a bit.
The Cauchy convergence criterion allows us to drop a finite number
of terms, and then capture everything after that point in a ball
of radius . As shrinks, all the terms in the
sequence are "squeezed togeher".
In the case, only successive terms must maintain
an distance. But as the example shows, you can steadily
plod along, keeping ball next to ball, to reach:
whose behaviour can do unexpected things depending on the choice of .