XII. Uniform Convergence #

Problems on uniform convergence of sequences of functions and on which properties of the functions in a sequence are inherited by the limit.

Problem 1 #

Recall the following definition from class:

Definition. A sequence of functions $\{f_n\}$ from a set $S$ to the real numbers is said to converge uniformly to a function $f$ iff for every $\epsilon > 0$, there is an integer $N$ such that $n \geq N$ implies that
$$\sup_{x \in S} |f(x) - f_n(x)| < \epsilon.$$

This is not a completely new notion. If the functions are also assumed to be continuous and $S$ is a compact metric space, then a sequence of functions $\{f_n\}$ converges to $f$ uniformly if and only if it converges to $f$ under the $\sup$ metric. A recurring theme in this week’s lectures was that a lot of reasonable properties of the functions in the sequence are passed to the limit function, but one has to be careful. The following exercises investigate the “limits” of this philosophy.

Suppose that the sequence of functions $\{f_n\}$ converges uniformly to the function $f$ and the sequence of functions $\{g_n\}$ converges uniformly to $g$.

(a) Show that $\{f_n + g_n\}$ converges uniformly to $f+g$.

(b) We begin with a definition.

Definition. A sequence of functions $\{f_n\}$ is bounded if there is a number $B$ such that $|f_n(x)| < B$ for all $x$ and all $n \in \mathbb{N}$.

Show that if $\{f_n\}$ and $\{g_n\}$ are bounded, then $\{f_n g_n\}$ converges uniformly to $fg$.

(c) Show that the boundedness restriction in the above statement is necessary. That is, find examples of $f_n$ and $g_n$ that converge uniformly to functions $f$ and $g$, but whose product does not converge uniformly to $fg$.

Extra Credit. Prove the Cauchy Criterion for Uniform Convergence:

Theorem. The sequence $\{f_n\}$ converges uniformly on $S$ iff $\forall \epsilon > 0, \; \exists N$ such that $n,m > N$ and $x \in S$ implies
$$|f_n(x) - f_m(x)| < \epsilon.$$

This is a crucial lemma which will be necessary to show that the space of continuous bounded functions on a metric space is a complete metric space when endowed with the uniform metric.

Solutions: Hint · Solution · Long solution

Problem 2 #

Suppose that $f_n, f : \mathbb{R} \to \mathbb{R}$ and define $f_n(x) = \frac{x}{1+n^2x^2}$. Show that the sequence $\{f_n\}$ converges uniformly to some function $f$, but

$$f'(0) \neq \lim f_n'(0).$$

In other words, derivatives are not preserved by uniform convergence. In class, we derived additional hypotheses which ensure this.

Solutions: Hint · Solution · Long solution

Problem 3 #

The following exercise forms the backbone for the so-called method of moments used in probability and statistics, although it has a variety of other applications. If $f$ is a function defined on $[0,1]$, its $n$th moment $M_n$ is defined as

$$M_n = \int_0^1 f(x)\, x^n \; dx.$$

Suppose that $f: [0,1] \to \mathbb{R}$ is continuous and that $M_n = 0$ for every non-negative integer $n$. Prove that $f$ must equal zero on $[0,1]$.

Hint: The hypothesis implies that the integral of the product of $f$ with any polynomial is zero. Use the Weierstrass Approximation Theorem to show that $\int_0^1 f^2(x) \; dx = 0$, from which you should be able to deduce that $f(x) = 0$.

Stray Observation: For those of you familiar with a bit of fancy linear algebra, this is morally very similar to showing that if the inner product of a vector with every element of a basis is zero, then the vector itself must be zero. But don’t use linear algebra for this problem; there are too many tricky details.

Solutions: Hint · Solution · Long solution