Question

Prove that :-

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

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

Answer 1

Answer:

Refer the attachment:)

Answer 2

Answer:

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