Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics II / 5

The sequence of functions $(f_n)$ converges uniformly to $f$ on $E$ ifThe same $N$ must work for every $x$. Equivalently, the supremum norm of $f_n-f$ tends to zero. This is stronger than pointwise convergence, where $N$ may depend on $x$.

The Uniform Cauchy criterion says that $(f_n:E\to \mathbb{R})$ converges uniformly on $E$ if and only ifFor necessity, use the triangle inequality through the uniform limit. For sufficiency, the condition makes $(f_n(x){)}_n$ a Cauchy sequence for each $x$, so completeness of the real numbers defines a pointwise limit $f(x)$. Apply the condition with $\epsilon /2$ and let $m\to \infty$ to obtain $|f_n(x)-f(x)|\le \epsilon /2<\epsilon$ uniformly in $x$.

Use uniform convergence to choose $N$ with $|f_n(x)-f_N(x)|\le 1$ for all $n\ge N$ and $x\in E$; this follows, for example, from the Uniform Cauchy criterion. Since each $f_j$ is a bounded function, put $M_j={\sup }_E|f_j|<\infty$ for $1\le j\le N$. Thenfor every $n$ and $x$. Thus $(f_n)$ is a uniformly bounded family of functions. The finite initial segment is essential: uniform convergence directly controls only a tail of a sequence.

Write $f_n\to f$ and $g_n\to g$ with uniform convergence. The triangle inequality givesHence $f_n+g_n\to f+g$ with uniform convergence.

If every $f_n,g_n$ is a bounded function, part iii gives constants $M_f,M_g$ bounding the two families. Their pointwise limits satisfy $|f|\le M_f$ and $|g|\le M_g$. ConsequentlyThus both sums and products have uniform convergence under the stated hypotheses. The product estimate is the uniform convergence product rule for bounded functions.

On $E=\mathbb{R}$, take $f_n(x)=x$ and $g_n(x)=1/n$. Then $f_n\to f(x)=x$ with uniform convergence, because there is no error, and $g_n\to 0$ with uniform convergence, with error $1/n$. Their products $f_n{g}_n=x/n$ converge pointwise to zero, but for every $n$ the choice $x=n$ gives $|f_n(x)g_n(x)|=1$. Thus the products do not converge uniformly. Here the first family consists of unbounded functions, exactly the missing hypothesis in the preceding estimate.