3.9 Uniform convergence

Analysis of functions of a single Page 1 / 3

We introduce now two different notions of the limit of a sequence of functions. Some definitions cover uniform convergence and pointwise convergence. The Weierstrass M-Test is stated and proven, as well the theorem of Abel.

We introduce now two different notions of the limit of a sequence of functions. Let $S$ be a set of complex numbers, and let ${f_{n}}$ be a sequence of complex-valued functions each having domain $S .$

We say that the sequence ${f_{n}}$ converges or converges pointwise to a function $f : S \to C$ if for every $x \in S$ and every $ϵ > 0$ there exists a natural number $N,$ depending on $x$ and $ϵ,$ such that for every $n \geq N,$ $| f_{n} (x) - f (x) | < ϵ .$ That is, equivalently, ${f_{n}}$ converges pointwise to $f$ if for every $x \in S$ the sequence ${f_{n} (x)}$ of numbers converges to the number $f (x) .$

We say that the sequence ${f_{n}}$ converges uniformly to a function $f$ if for every $ϵ > 0,$ there exists an $N,$ depending only on $ϵ,$ such that for every $n \geq N$ and every $x \in S,$ $| f_{n} (x) - f (x) | < ϵ .$

If ${u_{n}}$ is a sequence of functions defined on $S,$ we say that the infinite series $\sum u_{n}$ converges uniformly if the sequence ${S_{N} = \sum_{n = 0}^{N} u_{n}}$ of partial sums converges uniformly.

These two definitions of convergence of a sequence of functions differ in subtle ways. Study the word order in thedefinitions.

Prove that if a sequence ${f_{n}}$ of functions converges uniformly on a set $S$ to a function $f$ then it converges pointwise to $f .$
Let $S = (0, 1),$ and for each $n$ define $f_{n} (x) = x^{n} .$ Prove that ${f_{n}}$ converges pointwise to the zero function, but that ${f_{n}}$ does not converge uniformly to the zero function. Conclude that pointwise convergence does not imply uniform convergence. HINT: Suppose the sequence does converge uniformly.Take $ϵ = 1 / 2,$ let $N$ be a corresponding integer, and consider $x$ 's of the form $x = 1 - h$ for tiny $h$ 's.
Suppose the sequence ${f_{n}}$ converges uniformly to $f$ on $S,$ and the sequence ${g_{n}}$ converges uniformly to $g$ on $S .$ Prove that the sequence ${f_{n} + g_{n}}$ converges uniformly to $f + g$ on $S .$
Suppose ${f_{n}}$ converges uniformly to $f$ on $S,$ and let $c$ be a constant. Show that ${c f_{n}}$ converges uniformly to $c f$ on $S .$
Let $S = R,$ and set $f_{n} (x) = x + (1 / n) .$ Does ${f_{n}}$ converge uniformly on $S ?$ Does ${f_{n}^{2}}$ converge uniformly on $S ?$ What does this say about the limit of a product of uniformly convergent sequences versus the product of the limits?
Suppose $a$ and $b$ are nonnegative real numbers and that $| a - b | < ϵ^{2} .$ Prove that $| \sqrt{a} - \sqrt{b} | < 2 ϵ .$ HINT: Break this into cases, the first one being when both $\sqrt{a}$ and $\sqrt{b}$ are less than $ϵ .$
Suppose ${f_{n}}$ is a sequence of nonnegative real-valued functions that converges uniformly to $f$ on $S .$ Use part (f) to prove that the sequence ${\sqrt{f_{n}}}$ converges uniformly to $\sqrt{f} .$
For each positive integer $n,$ define $f_{n}$ on $(- 1, 1)$ by $f_{n} (x) = {| x |}^{1 + 1 / n} .$ Prove that the sequence ${f_{n}}$ converges uniformly on $(- 1, 1)$ to the function $f (x) = | x | .$ HINT: Let $ϵ > 0$ be given. Consider $| x |$ 's that are $< ϵ$ and $| x |$ 's that are $\geq ϵ .$ For $| x | < ϵ,$ show that $| f_{n} (x) - f (x) | < ϵ$ for all $n .$ For $| x | \geq ϵ,$ choose $N$ so that $| ϵ^{1 / n} - 1 | < ϵ .$ How?

Let ${f_{n}}$ be a sequence of functions on a set $S,$ let $f$ be a function on $S,$ and suppose that for each $n$ we have $| f (x) - f_{n} (x) | < 1 / n$ for all $x \in S .$ Prove that the sequence ${f_{n}}$ converges uniformly to $f .$

We give next four important theorems concerning uniform convergence. The first of these theorems is frequently used to prove that a givenfunction is continuous. The theorem asserts that if $f$ is the uniform limit of a sequence of continuous functions, then $f$ is itself continuous.

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Source: OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1

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