Re: What is the relation between Cauchy sequence and convergent sequence?

Hi Arturo,

"Depends on your definition of "trivial". Take as a metric space the
subspace of R given by [0,1] U [2,3] with the usual induced
metric. Then both [0,1] and [2,3] are open in R, both nontrivial, and
both complete. In this case, the two sets are clopen (both open and
closed), of course, so maybe you want to throw those into your
definition of "trivial.""

->[0, 1] and [2, 3] are closed in R, right?

Also, "closedness" and "completeness" looks awfully similar to me in
the metric space...

See the definition on Mardsen:

> 1. A set A contained in metric space M is closed iff for every sequence
> x_k contained in A that converges in M, the limit lies in A;

> 2. The space M is called complete iff every Cauchy sequence in M
> converges to a point in M.
> (In understanding this definition, I am thinking that we can also
> define complete sets -- the set A contained in metric space M is called
> complete iff every Cauchy sequence in A converges to a point in A.)

So we don't need both definitions, right?

.

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