Try the contrapositive: if the complement of E is NOT dense in R, then what can you say about E?
ok so if i go roughly like this would it be true:
Suppose E^c is not dense then there exist an interval I in R such that I is contained in E^c.
the lebesque outermeasure is denoterd by ð*. so ð*(E)=ð*(EI)+ð*(EI^c)= 0 + ð*(I^c) which is differenty than zero since I is an interval.