Please help me prove the following question:

1) Let E be a measurable subset of [0,1]. Prove that if both m(E) > 0 and m(E") > 0, then there exists a point p E [0, 1] such that for every Open neighborhood U of p in [0,1], m(EU)>0 and m(EcU) >0. (Hint: Put L0 = [0, i] and R0 = [51]. If either |EnL0|=0 and |EnR0|=0, 01' |Er1R0|=0 and |EnL0| :0, then p : % has the required preperty. If neither of these cases occurs...)