How would you show that a finite set has no cluster points?
If this is a topological space, this is wrong.
Think about where . It is easy to see that is a limit point of the set . For, the only neighborhood of is which contains a point of distinct from itself.
In a metric space though, think about the set and for any take and consider
x is a cluster point of A if every neighborhood of x contains some point of A other than x. Since A contains only a finite number of points, there can be only a finite number of points of A in any neighborhood of x. That means that, in any neighborhood of x, there is a point of A, other than x itself, which is closest to x. Look at a neighborhood of x with radius less than that least distance.