Is Every Convergent Sequence in a Closed Set a Cauchy Sequence?

[kathrynag]

Homework Statement

A set F\subseteqR is closed iff every Cauchy sequence contained in F has a limit that is also an element of F.

Homework Equations

The Attempt at a Solution

Let F be closed. Then F contains its limit points.
This means x=lima_{n} are elements of F.

 

[kathrynag]

I guess the formal proof is getting me:
Every closed set contains all of its limit points. Then there is a a neighborhood(x) such that the intersection of F is not equal to {x}. I guess I don't see how to get to the Cauchy sequence from there.

Other direction: Let an be cauchy sequence such that an has a limit contained in F.
an is a Cauchy sequence if abs value(an-am)<epsilon
A limit exists for an if abs value(an-a)<epsilon
we can rewrite an-am as (an-a)+(a-am)
Then by the triangle inequality, abs value(an-am)<abs value(an-a)+abs value(a-am)
I'm stuck going from there to the set being closed

 

[eileen6a]

i think every convergent sequence in F is a cauchy sequence.