Help with details in an example showing the function $f(x)=\frac{1}{x^2}$ is not uniformly continuous

Question

Here is a portion of an example in Ross' Elementary Analysis. Definition 19.1 is the definition of uniform continuity.

What is the rationale for choosing $\frac{\delta}{2}$?

Answer 1

To show (1) we need two numbers $x,y$ such that $|x-y|<\delta$. Instead of choosing $x$ and $y$ separately, it's easier to decide that $y$ will be $x+$(something small) and focus on choosing $x$. The numbers $x$ and $x+\delta$ would not work because they do not satisfy the strict inequality $\dots<\delta$. So we use $\delta/2$ instead, because $\delta/2$ is the world's favorite number between $0$ and $\delta$.

Answer 2

Uniform continous functions transform Cauchy sequences into Cauchy sequences. This function transform the Cauchy sequence $u_n=1/n$ into the unbounded sequence $f(u_n)=n^2$ which is in turn not a Cauchy sequence.