Mathematics III - Foundations of Analysis - Summer 24

I. Consider following examples.  Do computations according to the instructions, Observe your results and take notes.  Your possible questions should be clarified after this course.

1. $f_n(x)=\dfrac{x}{x+n}$, $n=1,2,...$, $x\in{\mathbb Z}_+$.  Fix $n$ compute $\displaystyle\lim_{n\to\infty}\lim_{x\to\infty}f_n(x)$ and $\displaystyle\lim_{x\to\infty}\lim_{n\to\infty}f_n(x)$.  Are they equal?
2. $f_n(x)=\dfrac{x^2}{(x^2+1)^n}$, $n=1,2,...$.  Show that $\displaystyle\sum_{n=0}^\infty f_n(x)$ converges, denote the limit $f(x)$.  Is $f(x)$ continuous?  Are $f_n(x)$'s continuous?
3. $f_n(x)=\dfrac{nx}{(nx+1)^3}$, for $0\le x$.  Compute $\displaystyle\lim_{n\to\infty}f_n(x)$, call it $f(x)$, $\displaystyle\lim_{n\to\infty}f_n'(x)$.  Does $\{f_n'\}$ converge to $f'$?
4. $f_n(x)=\begin{cases}2n&\text{if }x\in[1/2n,1/n]\\0&\text{otherise}\end{cases}$ on $[0,1]$.  Compute the limit of $f_n$ for fixed $x\in[0,1]$, call the limit function $f$.  Integrate $f(x)$ from $0$ to $1$.  Compute also integral of $f_n$ from $0$ to $1$ for fixed $n$, then take the limit.  What is your observation?

These examples are discusson on when two limit processes can be interchanged.  More precisely, we ask if the following things are true:

• $\displaystyle\lim_{n\to\infty}\lim_{x\to a}f_n(x)=\lim_{x\to a}\lim_{n\to\infty}f(x)$
• $\displaystyle\lim_{n\to\infty}f_n'=D(\lim_{n\to\infty}f_n)$
• $\displaystyle\lim_{n\to\infty}\int_a^bf_n=\int_a^b\lim_{n\to\infty}f_n$
• $\displaystyle\sum_{n=1}^\infty f_n'(x)=D(\sum_{n=1}^\infty f_n(x))$
• $\displaystyle\sum_{n=1}^\infty\int_a^b f_n=\int_a^b\sum_{n=1}^\infty f_n$
• $\dfrac{d}{dx}\int_a^bf(x,y)dy=\int_a^b\frac{\partial }{\partial x}f(x,y)dy$

The above examples show that notion of the limit of $f_n(x)$ for fixed $x$ in a metrc space (called pointwise convergence) is too weak to guarantee the limit interchange mentioned here:  In words, it does not preserve continuity, it does not preserve limits, it does not preserve integrals. This motivates a new notion of convergence, which is called uniform convergence. 

II. Now we discuss these two notions of convergence of a sequence of functions.  We should compare uniform with pointwise convergence:

• For pointwise convergence we could first fix a value for $x$ and then choose $N$. Consequently, $N$ depends on both $\varepsilon$  and $x$.

• For uniform convergence $f_n(x)$ must be uniformly close to $f(x)$ for all $x$ in the domain. Thus $N$ only depends on $\varepsilon$ but not on $x$.

Let's illustrate the difference between them graphically:

Figure below left:  (pointwise convergence) First fix a value $x_0$. Then we choose an arbitrary neighborhood around $f(x_0)$, which corresponds to a vertical interval centered at $f(x_0)$. Finally we pick $N$ so that $f_n(x_0)$ intersects the vertical line $x_0$ inside the interval $(f(x_0)-\varepsilon,f(x_0)+\varepsilon)$

Figure below right: (uniform convergence) We draw an $\varepsilon$-neighborhood around the entire limit function $f(x)$, which results in an "$\varepsilon$-strip" with $f(x)$ in the middle. Now we pick $N$ so that $f_n(x)$ is completely inside that strip for all $x$ in the domain.

As an immediate consequence of the defition we have the following useful criterion

Theorem.  Suppose $\displaystyle\lim_{n\to\infty}=f(x)$ ($x\in E$).  Let $\displaystyle M_n=\sup_{x\in E}|f_n(x)-f(x)|$.  Then $f_n\to f$ uniformly on $E$ if and only if $M_n\to 0$ as $n\to\infty$.

Note that constructing a counterexample to show a statement is not true is part of mathematical training.