Lebesgue integrals measure of set

Question

If a<c<b and E is a measurable subset of [a,b] ,show that E intersection with [a,c] is a measurable subset of [a,c]. I have try to prove using definition of measurable set but I am not sure it is correct. My ideas is since given set is subset of [a,c] and E is measurable then E intersection with [a,c] is contain in E hence outer measure of E intersection with [a,c] is less than outer measure of E , similarly inner measure of E intersection with [a,c] is less than inner measure of E. Now subtract this two inequalities we get outer measure equal to inner measure for set E intersection with [a,c]. Please review the proof .