Let f_n : [0,1] → [0,1] be a sequence of Riemann integrable functions, and f : [0, 1] → [0, 1] be a function so that for each k there is N_k so that supremum_(1/k<x≤1) of |f_n(x) − f(x)| < 1/k , for n ≥ N_k . Prove that f is Riemann integrable and ∫ f(x) dx = lim_n→∞ ∫ f_n(x) dx(adsbygoogle = window.adsbygoogle || []).push({});

I am really lost, so any kind of help would be greatly appreciated.

The part that confuses me the most is "for each k there is N_k so that supremum_(1/k<x≤1) of |f_n(x) − f(x)| < 1/k , for n ≥ N_k"

It seems that the sequence of functions are integrable on [0,1] but there is a problem at 0 because of sup (1/k<x ≤1). This means that we need to split the integral into to parts, one from 0 to 1/k and the other one from 1/k to 1. In the end we would see that having a point on the interval that is problematic does not make the function not Riemann integrable.

Am I correct?

Once I split the integral should I just integrate both of them using the definition (Riemann integral)?

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# Riemann integrable sequences of functions

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