analysis with ....

hello sir~

f:[0,1] -> R is continuous.

g_n(x) = f(x^n), x in [0,1] (n in N)

show that integral{0 to 1} g_n converge to f(0).

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i think....
it should seem that this is related to
Lebesque's convergence theorem(bouned convergence theorem)

so,
let h(x) = lim g_n(x)
then, h(x) = lim g_n(x) = f(0) , 0<= x< 1
if x= 1, h(x) = lim g_n(x) = f(1)

so, h(x) = f(0) ,x in [0,1)
= f(1) , x =1.

and |g_n(x)| = |f(x_n)| < M by continuous and [0,1].

by Lebesque's convergence theorem,
lim integral{0 to 1} g_n(x) dx
= int{0 to 1} lim g_n(x) dx
= int{0 to 1} h(x) dx
= lim{a->1-} int{0 to a} h(x) dx
= lim{a->1-} int{0 to a} f(0) dx
= lim{a->1-} f(0)*a
= f(0)

is this no problem ?


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