If D⊂R(reals) , prove that closure(D) = DUD' is bounded?

May 2021 2 72 Report

If D⊂R(reals) , prove that closure(D) = DUD' is bounded?

If D⊂R(reals) , prove that closure(D) = DUD' is bounded where D' is the set of all accumulation points of D.

Update:

Oh I forgot to mention that D is bounded.

Let D be a bounded subset of R. There exists an M > 0 such that D is a subset of

(-M,M). Then closure(D) is a subset of [-M,M], which is a subset of (-(M+1),M+1). Therefore, closure(D) is bounded.

IMO in the form written by the asker, the statement is false,

Consider rationals Q⊂R

closure(Q) = R

which is not bounded

Example #4