Accumulation points of a countable set in [0,1]

Let A be a countable subset of [0,1]. Denote the set of accumulation points of A by A'. Can the set of accumulation points of A' be nonempty (i.e. can we have that A'' is nonempty)?

• Hint: What is the "simplest" countable subset of $[0,1]$? – njguliyev Oct 28 '13 at 21:09
HINT: Take $A=\Bbb Q\cap[0,1]$.