2007-24.md

course	course_year	question_number	tags	title	year
Analysis II	IB	24	IB 2007 Analysis II	4.I.3H	2007

Define uniform convergence for a sequence $f_{1}, f_{2}, \ldots$ of real-valued functions on the interval $(0,1)$.

For each of the following sequences of functions on $(0,1)$, find the pointwise limit function. Which of these sequences converge uniformly on $(0,1)$ ?

(i) $f_{n}(x)=\log \left(x+\frac{1}{n}\right)$,

(ii) $f_{n}(x)=\cos \left(\frac{x}{n}\right)$.

Justify your answers.