Question

2. For each natural numbern and each number x in [0, 1), define f,(x) nxe Prove that the sequence (f: [0, 1] R} converges pointwise to the constant function 0, but that the sequence of integrals (of,) does not converge to 0. Does this contradict Theorem 9.18?

I'm not good at front of advanced math because I'm in the middle of the class. If you show me a rigorously detailed proof, I'd like to ask a additional question probably...

Answer 1

Question For each natural number n and each number x in [0,1] define f_n(x) = nx e^-n x^2 prove that the sequence {f_n: [0,1] rightarrow R} converges point wise to the constant function 0, but that the sequence of integrals integrals^1_0 f_n does not converge to 0 Does this contradict the theorem Thm: suppose that: f_n: [a,b] rightarrow R} is a sequence of integral fun that converges uniformly to the function f: [a,b] rightarrow R is also integrable , moreover Lim_n rightarrow infinity [integral^b_a f_n] = integral^b_a f Here f(x) = Lim_n rightarrow infinity f_n(x) = Lim_n rightarrow infinity nx e^- n x^2 = Lim_n rightarrow infinity nx/e^nx^2 = 0 [As exponential goes much to infinity a than nx] thus from 1 the sequence {f_n} converges to 0 on any interval [0,k] , k > 0 here k = 1 that is [0,1] Let us suppose, if possible , the sequence {f_n} converges uniformly on [0,1]