Prove that :-

If [tex]{\bf \{ f_{n} \}}[/tex] be a sequence of real functions which converge uniformly to [tex]\bf f[/tex] on the interval [tex]{\bf a \leqslant x \leqslant b}[/tex] also suppose that each function [tex]{\bf f_{n} ( n = 1 , 2 , 3 \cdots \cdots )}[/tex] is continuous on [tex]{\bf a \leqslant x \leqslant b}[/tex] . Then for every [tex] \alpha [/tex] and [tex] \beta [/tex] such that [tex]{\bf a \leqslant \alpha < \beta \leqslant b}[/tex] . Then ;

[tex]{\boxed{\displaystyle \bf \lim_{n \to \infty} \displaystyle \bf \int_{\alpha}^{\beta} f_{n} (x ) dx = \displaystyle \bf \int_{\alpha}^{\beta} \displaystyle \lim_{n \to \infty} \bf f_{n} (x ) dx}}[/tex] ​