Countable bounded set

Homework Statement

given a set A(subset of R(reals)) is bounded.and for all x belongs to R there exists epsilon(eps) such that {(x-eps,x+eps) intersection A} is countable..to prove A is countable

The Attempt at a Solution

...bdd set in R is totally bounded...but iam not finding the way how to cover A by epsilon cover(has at most countable elements)