Sequence of measurable functions with |f_n|^2<g

Question

Suppose {f_n} is a sequence of measurable functions on [0,1] such that limit(as n goes to infinity) integral over [0,1] |f_n|=0 and that there is an integrable function g on [0,1] such that |f_n|^2 is less than or equal to g. Prove that limit(as n goes to infinity) integral over [0,1] |f_n|^2=0