Given A is a subset of RdSolutionSuppose x0 is one of the li

Given A is a subset of R^d.

Solution

Suppose x0 is one of the limit points of A, then for any given epsilon, we can find x in A\' such that |x-x0|<epsilon/2.

Now x becomes another limit point of A abd hence there is one x1 in A\' such that |x1-x|<epsilon/2

Hence |x1-x0|<epsilon.

Hence x0 is a limit point of A and is contained in A\'.

But A\' contains all limit points.

It follows that

A\' = set of limit points of closure of A