Lenesgue integrals

Prove or disprove the following: If f is in L^1[0,1], then limit the integral over [0,1] of x^n*f = 0 as n goes to infinity. I saw a similar example asking to prove that the integral from 0 to 1 of x^2n f(x) dx = 0, and they used algebra of functions generated by {1,x^2}, but we haven't talked about that, so please when you prove or disprove, use basic things we know about Lebesgue measures and integrals, since all integrals here are with respect to Lebesgue measure. If you can't do it this way, then please don't answer my Q. Thanks in advance.

Converge uniformly

It is explain in attach. We are usining the book Methods of Real Analysis by Richard R. Goldberg. Please can you explain the problems step by step and don't use a theorem that is not in the respective chapter.

Sequence

(See attached file for full problem description with proper symbols) --- 9.4-6 Let be a sequence of continuous real-valued functions that converges uniformly on the closed bounded interval [a, b]. For each let Show that converges uniformly on [a,b]. (Hint: Use 9.2F) Theorem 9.2F; Let be a sequence of real-valued functions on a set E. Then is uniformly convergent on E ( to some function) if and only if given there exists such that ---

Sequence of cont. function

(See attached file for full problem description with proper symbols) --- 9.4-8 Let be a sequence of continuous functions [0,1] that converges uniformly. a) Show that there exists M>0 such that b)Does the result in part (a) hold if uniform convergence is replaced by pointwise convergence? ---

limit

(See attached file for full problem description with equation and proper symbols) --- 9.2-10 If be a sequence of functions that converges uniformly to the continuous function , prove that ---