Suppose that h_n is a sequence of non-negative integrable functions on [a,b], su

Question

Suppose that h_n is a sequence of non-negative integrable functions on [a,b], such that the limit of the integral (from a to b) of h_n(x)*dx = 0 (as n goes to infinity).
(a)Show that if f is integrable on [a,b], then the limit of the integral (from a to b) of f(x)*h_n(x)*dx = 0 (as n goes to infinity).
(b)Use this result to show that if f is integrable on [0,1], then the integral (from 0 to 1) of x^(n)*f(x)*dx = 0

I broke up the last part from the first, but i'm hoping someone

can help explain the first part?

Explanation / Answer

we can easily prove that as a) isalready proved

we know that f(x)*h_n(x)*dx = 0

=>here substitute [a,b] as [0,1] and we can find that it approaches to 0 when n tends to infinity and integrate from [0,1] so now the value of lim becomes 0 at 1

we can also see that at 0 the value is exactly 0

hence value of lim = 0 - 0 =>0

hence proved

qed

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