Sachchidanand Prasad

09-01-2023

Problem: Find a continuous function $f:I\to \mathbb{R} $ and a Cauchy sequence $\left( x_n \right) $ such that $\left( f\left( x_n \right) \right) $ is not Cauchy.
Solution: Consider the function \[ f : (0,\infty ) \to \mathbb{R} , \quad f(x) = \frac{1}{x}. \] It is clear that the function is continuous on $(0, \infty )$. We take a sequence $\left( \frac{1}{n} \right) $, which is convergent in $\mathbb{R} $ and hence Cauchy. The image \[ f\left( \frac{1}{n} \right) = n, \] is not Cauchy.