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## Homework Statement

L is the set of limit point of A in the real space, prove that L is closed.

## Homework Equations

## The Attempt at a Solution

L may or may not have limit points. If L does not have limit points, then it's obviously closed.

If L has limit points, the let l be a limit points of L. => exists x in L such that x is in the [tex]\frac{\epsilon}{2}[/tex] neighborhood of l and x[tex]\neq[/tex]l

x is a limit point of A => exists a in A such that a is in the [tex]\frac{\epsilon}{2}[/tex] neighborhood of x and a[tex]\neq[/tex]x

a is in the [tex]\epsilon[/tex] neighborhood of l. The problem is, how do i show that a[tex]\neq[/tex]l?

Thanks!