Why Are All Points in [0,1] Cluster Points of the Interval (0,1)?

[happyg1]

cluster point confusion...

Fog Fog Fog...
Ok,Here's the question:
Let A denote the open interval (0,1).Show that the set of Cluster points of A in $$R^1$$ is [0,1].
Our textbook sez that $$R^1$$ is defined as the absolute value metric, i.e. $$\rho$$(x,y)=|x-y|
OK
So I know (and have proven) that (0,1) is uncountable and that there are infinitely many points between any 2 points in (0,1). It is easy for me to understand that the definition of a cluster point
we which we have as:
"let M,$$\rho$$ be a metric space and suppose A $$\subset$$M. The point a $$\in$$M is called a cluster point of A in M if, for every h>0, there exists a point x$$\in$$A such that 0,$$\rho$$(x,a)<h."
is fulfilled...no matter what h I pick, I will always be able to find some x...
I just don't know where to start to write it down in a fashion that my prof. would accept. He's VERY picky and I'm VERY tired.
Give me some nudges, please...
BTW...If ANY of Ya'll have kids, WATCH OUT for the rotavirus...wash your hands A LOT. Both of my littles have been hospitalized for dehydration (it's gnarly stomach flu...)
Thanks,
CC

 

[matt grime]

It's just the closure.

If you need to use the definition as given, and you already know that the answer is [0,1] just do it: if x is not in [0,1] show it is not a cluster point, and if x is in [0,1] show it is a cluster point, obviously all points in (0,1) are cluster points, so it only remains to show that 1 and 0 are cluster points which is exactly as hard as knowing that 1/n tends to 0