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 ...

https://www.dpmms.cam.ac.uk/~wtg10/autoanalysis.html

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.

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

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

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

$latex \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:

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

Take arbitrary a as below:

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


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

$latex \iff \{a_n\} \:\: diverges$

Exercise:
Prove:
1-1+1-1+1...
=1, or
=0, or
= 1/2 (Leibniz said 50% -1 50% 0) ?