| course |
course_year |
question_number |
tags |
title |
year |
Analysis II |
IB |
5 |
|
Paper 3, Section II, F |
2018 |
(a) Let $A \subset \mathbb{R}^{m}$ and let $f, f_{n}: A \rightarrow \mathbb{R}$ be functions for $n=1,2,3, \ldots$ What does it mean to say that the sequence $\left(f_{n}\right)$ converges uniformly to $f$ on $A$ ? What does it mean to say that $f$ is uniformly continuous?
(b) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a uniformly continuous function. Determine whether each of the following statements is true or false. Give reasons for your answers.
(i) If $f_{n}(x)=f(x+1 / n)$ for each $n=1,2,3, \ldots$ and each $x \in \mathbb{R}$, then $f_{n} \rightarrow f$ uniformly on $\mathbb{R}$.
(ii) If $g_{n}(x)=(f(x+1 / n))^{2}$ for each $n=1,2,3, \ldots$ and each $x \in \mathbb{R}$, then $g_{n} \rightarrow(f)^{2}$ uniformly on $\mathbb{R}$.
(c) Let $A$ be a closed, bounded subset of $\mathbb{R}^{m}$. For each $n=1,2,3, \ldots$, let $g_{n}: A \rightarrow \mathbb{R}$ be a continuous function such that $\left(g_{n}(x)\right)$ is a decreasing sequence for each $x \in A$. If $\delta \in \mathbb{R}$ is such that for each $n$ there is $x_{n} \in A$ with $g_{n}\left(x_{n}\right) \geqslant \delta$, show that there is $x_{0} \in A$ such that $\lim {n \rightarrow \infty} g{n}\left(x_{0}\right) \geqslant \delta$.
Deduce the following: If $f_{n}: A \rightarrow \mathbb{R}$ is a continuous function for each $n=1,2,3, \ldots$ such that $\left(f_{n}(x)\right)$ is a decreasing sequence for each $x \in A$, and if the pointwise limit of $\left(f_{n}\right)$ is a continuous function $f: A \rightarrow \mathbb{R}$, then $f_{n} \rightarrow f$ uniformly on $A$.