2001-19.md

course	course_year	question_number	tags	title	year
Analysis II	IB	19	IB 2001 Analysis II	4.II.10A	2001

Let $\left{f_{n}\right}$ be a pointwise convergent sequence of real-valued functions on a closed interval $[a, b]$. Prove that, if for every $x \in[a, b]$, the sequence $\left{f_{n}(x)\right}$ is monotonic in $n$, and if all the functions $f_{n}, n=1,2, \ldots$, and $f=\lim f_{n}$ are continuous, then $f_{n} \rightarrow f$ uniformly on $[a, b]$.

By considering a suitable sequence of functions $\left{f_{n}\right}$ on $[0,1)$, show that if the interval is not closed but all other conditions hold, the conclusion of the theorem may fail.