Problem:
$f:(0,\infty )\rightarrow \mathbb{R}$ defined as: $f(x)=x^{2}$
Can anyone show me how to prove that $f$ transforms Cauchy sequences of elements of $(0,\infty )$ into Cauchy sequences, but $f$ is not uniformly continuous? The purpose of this problem is to show how essential the boundedness of the set on which $f$ is defined is.
$f$ is definitely not uniformly continuous because for the two sequences: $\left \{ x_{n} \right \},\left \{ y_{n} \right \}$ defined by: $x_{n}=n+\frac{1}{n}$ and $y_{n}=n$. We have: $\left | x_{n}-y_{n} \right | \to 0$ as $n \to \infty $, but $\left | f(x_{n})-f(y_{n}) \right | \to 2$ as $(n \to \infty )$. Can anyone, please, show me how to prove that $f$ transforms Cauchy sequences of elements of $(0,\infty )$ into Cauchy sequences? Thanks