2017-1.md

course	course_year	question_number	tags	title	year
Analysis II	IB	1	IB 2017 Analysis II	Paper 2, Section I, G	2017

Let $X \subset \mathbb{R}$. What does it mean to say that a sequence of real-valued functions on $X$ is uniformly convergent?

Let $f, f_{n}(n \geqslant 1): \mathbb{R} \rightarrow \mathbb{R}$ be functions.

(a) Show that if each $f_{n}$ is continuous, and $\left(f_{n}\right)$ converges uniformly on $\mathbb{R}$ to $f$, then $f$ is also continuous.

(b) Suppose that, for every $M>0,\left(f_{n}\right)$ converges uniformly on $[-M, M]$. Need $\left(f_{n}\right)$ converge uniformly on $\mathbb{R}$ ? Justify your answer.