Definition: Suppose that S is a set of real numbers and that x is any real number. We say that the number x is a limit point of the set S if for every number $\displaystyle \delta>0$ we have $\displaystyle (x-\delta, x+\delta)\cap S$ \ $\displaystyle \{x\}$ is not equal to $\displaystyle \emptyset$. The set of limit points of a given set S is written as L(S).

Problem: Given any set S of real numbers, prove that the set L(S) must be closed.