Question Number 55642 by gunawan last updated on 01/Mar/19
![Prove the following statements: If for every n , f_n form ascend function and {f_n } uniform convergences to f at [a, b], then lim_(n→∞) ∫_a ^b f_n (x) dx →∫_a ^b f(x) dx](https://www.tinkutara.com/question/Q55642.png)
$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{following}\:\mathrm{statements}: \\ $$$$\mathrm{If}\:\mathrm{for}\:\mathrm{every}\:{n}\:,\:{f}_{{n}} \:\mathrm{form}\:\mathrm{ascend}\:\mathrm{function} \\ $$$$\mathrm{and}\:\left\{{f}_{{n}} \right\}\:\mathrm{uniform}\:\mathrm{convergences} \\ $$$$\mathrm{to}\:{f}\:\mathrm{at}\:\left[{a},\:{b}\right],\:\mathrm{then} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\int_{{a}} ^{{b}} {f}_{{n}} \left({x}\right)\:{dx}\:\rightarrow\int_{{a}} ^{{b}} {f}\left({x}\right)\:{dx} \\ $$