compactness help

[new]

(X,p) is a metrice space.
A is a subset of X.
Every sequence in A has a convergent subsequence.

-What is an example that shows that A is not necessarily compact?
-If we assume further that every cluster pt of A is an interior pt of A.
Is A compact then? (It seems so.) Can we give example of a metric space and a nonempty subset of A for this?

Many thanks.

[HallsofIvy]

(X,p) is a metrice space.
A is a subset of X.
Every sequence in A has a convergent subsequence.

-What is an example that shows that A is not necessarily compact?
-If we assume further that every cluster pt of A is an interior pt of A.
Is A compact then? (It seems so.) Can we give example of a metric space and a nonempty subset of A for this?

Many thanks.
First, what is your definition of "compact" (there are several equivalent definitions)?

Since the second part of the equation says "If we assume futher that every cluster pt of A is an interior pt of A" you might want to look for a counter example for the first part amoung sets where that is NOT true! Notice that saying "Every sequence in A has a convergent subsequence" does NOT necessarily mean that every sequence converges to something IN A.