Prove D U D' is bounded

Argonlight

The homework question is this:
Prove If D is a bounded subset of R then D bar = D U D’ is also bounded where D’ is the set of accumulation points of D.

What is a general outline of a proof?

joeboo

It suffices to show that D' is bounded, as the union of 2 bounded sets is bounded.

If D is bounded, then it is contained in some finite interval, ( - N, N ). If D' is not bounded, then we can find an element, x, of D' outside of ( -N, N ), and taking a suitable neighborhood around x ( of radius less than |x| - N ), we see that it is disjoint from D ( as it is disjoint from ( -N, N ) ). Therefore, x is not an accumulation point of D. Contradiction