Show that $f(x)=1/x^2$ is not uniformly continuous one the set $(0,1]$
Using the Sequential Criterion for Nonuniform Continuity - which states that
a function $f:A \rightarrow $ R fails to be uniformly continuous on A iff there exists a particular $\epsilon_0$>0 and two sequences ($x_n$) and ($y_n$) in A, satisfying $|x_n -y_n| \rightarrow 0$, but $|f(x_n) - f(y_n)|\ge \epsilon_0$
I would say:
Take ($x_n$) = $\frac{1}{n}$ and ($y_n$)=$\frac{1}{n^2}$. Obviously $|\frac{1}{n} - \frac{1}{n^2}| \rightarrow 0$, but $|f(x_n) - f(y_n)| = |n^2 - n^4| \ge 12 $, for example for n $\ge$ 2