This is a problem from a previous year's proficiency exam in my Master's program. I am taking this exam later this month. 1. The problem statement, all variables and given/known data Let f be a continuous function of the interval [0,1]. (1) If [tex]\int_0^1 f(x)^2 dx = 0[/tex] prove f(x)=0 for all x in [0,1]. (2) If [tex]\int_0^1 f(x) x^n dx = 0[/tex] (n=0,1,2...), prove f(x)=0 for all x in [0,1]. 2. Relevant equations There is a hint for the second part. "Use the Weierstrass Theorem: the space of polynomials is dense in the space of continuous functions." 3. The attempt at a solution I believe I have the first part pretty well figured out. Without going into all the gory details, the idea is that we assume there exist a c in [0,1] such that f(c) is not 0. Then by continuity there is an interval around c in which f(x) is positive (bounded away from 0). Thus, since f(x)2 is nonnegative on the whole interval and positive on the interval around c, the integral is positive, but this is a contradiction. I am at a loss for the second part. It seems to me that it should be a logical step from the first part but I cannot seem to make sense of it. Also, the hint should be a big help but I don't see how. Anything to put me on the right track is appreciated.