I just want a clear proof for this theorem.
Theorem. The closure of a set S in R^n is the set of all limits of converging sequences of points from S.
The closure of S in R^n is S unioned with the set of limit points of S, i.e. points x such that any neighborhood of x contains some point of S other than x itself. The exercise is to show that this condition on x is equivalent to being the limit of a convergent sequence of points in S.
Hint: every neighborhood of a point contains an epsilon-ball.