Prove that x is an accumulation point of a set
A in a metric space X if and only if for each r > 0 the set B(x, r) \ A is infinite.
Im findinding it hard to understand what is an accumulation point, I just dont know how to start this proof.
Prove that x is an accumulation point of a set
A in a metric space X if and only if for each r > 0 the set B(x, r) \ A is infinite.
Im findinding it hard to understand what is an accumulation point, I just dont know how to start this proof.
p is an accumulation point of a set, A, if and only if there exist other points of A arbitrarily close to p. For example, in the set (0, 1)U {2}, the open interval from 0 to 1 together with the single point 2, the set of accumulation points is precisely [0, 1]. 2 is NOT an accumulation point because there are no other points of the set within distance 1 of the point.
Do i do something like this..
Proof By Contradiction:
Suppose x is an accumulation point of A, and that B(x,r)∩A is finite......
I don't really know where to go from here, what is meant by accumulation point, I understand the concept... But just can't write it down mathematically...