Analysis by Timothy Gowers

Why easy analysis problems are easy
by Timothy Gowers (UK, Fields Medal 1998)

Timothy Gowers is teaching in Cambridge, he wrote the thick volume of “Princeton Math Encyclopedia.”

He is a very good mathematician, who likes to explain simple fundamental Math questions (like why 2+2=4, multiplication is commutative,…), in the process making abstract math simple to understand.

If you have recently met epsilons and deltas for the first time, then you may find the problems you are asked to solve on examples sheets very hard. On the other hand, you will notice that your lecturers, supervisors etc. do not find them hard at all. Why is this? ” Read on …

Below is my attempt to rewrite the Example 1 with Latex epsilon-delta notation for easy reading.

Example 1.

I wish to prove that the sequence (1,0,1,0,1,0,…) does not converge.

\text{Let me set the sequence }   \{a_n\} \text{ to be:}

\{a_n\}=  \begin{cases}  1, & \text{if }n \text{ is odd} \\  0, & \text{if }n\text{ is even}  \end{cases}

\Large\text{ Then the statement that }   \{a_n\} \Large\text{ converges to } a \Large\text{ can be written: }

\exists a, \forall \varepsilon >0 ,\:\:\exists N ,\:\:\forall n > N , \:\:|a_n - a| < \varepsilon

For divergence, we want to write the negation of the above as:

\boxed{\forall a,\: \exists \varepsilon >0,  \:\:\forall N, \:\:\exists  n > N, \:\:|a_n-a| \geq \varepsilon}

Take arbitrary a as below:

a_n = 1  \text{ if n is odd, choose }a < 1/2
a_n = 0 \text{ if n is even, choose }a \geq 1/2

\text {Let } \varepsilon = \frac {1}{2}
For either case whether n is even or odd,
\forall N, \:\:\exists  n > N, \:\: |a_n- a| \geq \frac{1}{2}

\iff \{a_n\} \:\: diverges

=1, or
=0, or
= 1/2 (Leibniz said 50% -1 50% 0) ?

The Princeton Companion to Mathematics

εδ Confusion in Limit & Continuity

1. Basic:
|y|= 0 or > 0 for all y

2. Limit: \displaystyle\lim_{x\to a}f(x) = L ; x≠a
|x-a|≠0 and always >0
\displaystyle\lim_{x\to a}f(x) = L
For all ε >0, there exists δ >0 such that
\implies |f(x)-L|< \epsilon

3. Continuity: f(x) continuous at x=a
Case x=a: |x-a|=0
=> |f(a)-f(a)|= 0 <ε (automatically)
So by default we can remove (x=a) case.

Also from 1) it is understood: |x-a|>0
Hence suffice to write only:

f(x) is continuous at point x = a
For all ε >0, there exists δ >0 such that
\implies |f(x)-f(a)|< \epsilon

Limit: ε-δ Analysis

For x->0, find limit L of

f(x)= (x³+5x)/x

1) guess L:

f(x)= x(x²+5)/x= x²+5

=> L= 5  when x->0

2) epsilon-delta Proof: find δ in function of ε such that:

|f(x)-5| < ε

|(x²+5)-5| <ε

|x|< √ε

Choose δ=√ε

For all ε, there is δ=√ε such that |x-0|< δ =>|f(x)-5|< ε

If ε=0.5, δ=√0.5=0.25

Rigorous Calculus: ε-δ Analysis

Rigorous Analysis epsilon-delta (ε-δ)
Cauchy gave epsilon-delta the rigor to Analysis, Weierstrass ‘arithmatized‘ it to become the standard language of modern analysis.

1) Limit was first defined by Cauchy in “Analyse Algébrique” (1821)

2) Cauchy repeatedly used ‘Limit’ in the book Chapter 3 “Résumé des Leçons sur le Calcul infinitésimal” (1823) for ‘derivative’ of f as the limit of

\frac{f(x+i)-f(x)}{i}  when i -> 0

3) He introduced ε-δ in Chapter 7 to prove ‘Mean Value Theorem‘: Denote by (ε , δ) 2 small numbers, such that 0< i ≤ δ , and for all x between (x+i) and x,

f ‘(x)- ε < \frac{f(x+i)-f(x)}{i} < f'(x)+ ε

4) These ε-δ Cauchy’s proof method became the standard definition of Limit of Function in Analysis.

5) They are notorious for causing widespread discomfort among future math students. In fact, when it was first introduced by Cauchy in the Ecole Polytechnique Lecture, the French Napoleon top students booed at him and Cauchy received warning from the school.

Note 1: From the textbook ‘Calculus‘ (1980, USA):
If can’t understand the ‘ε-δ’ definition, just memorize it like a poem – at least better than saying it wrongly.
E.g. “For all ε>0, there exists δ>0, …

Note 2: George Polya: “The students are not trained in ‘ε-δ’, teaching them Calculus is like dropping these rules from the sky...”