2006-20.md

course	course_year	question_number	tags	title	year
Analysis II	IB	20	IB 2006 Analysis II	2.I.3F	2006

Define uniform convergence for a sequence $f_{1}, f_{2}, \ldots$ of real-valued functions on an interval in $\mathbf{R}$. If $\left(f_{n}\right)$ is a sequence of continuous functions converging uniformly to a (necessarily continuous) function $f$ on a closed interval $[a, b]$, show that

$$\int_{a}^{b} f_{n}(x) d x \rightarrow \int_{a}^{b} f(x) d x$$

as $n \rightarrow \infty$.

Which of the following sequences of functions $f_{1}, f_{2}, \ldots$ converges uniformly on the open interval $(0,1)$ ? Justify your answers.

(i) $f_{n}(x)=1 /(n x)$;

(ii) $f_{n}(x)=e^{-x / n}$.