Answer to Question #280117 in Real Analysis for abc

Question

Let 𝑥𝑘 be a Cauchy sequence and 𝑓 be a continuous function, show that 𝑓(𝑥𝑘) is also a Cauchy sequence. 

Answer 1

Solution:

It is false.

Example:

Take

$\begin{gathered} X=(0,+\infty) \\ x_{k}=\frac{1}{k} \\ f: x \mapsto \frac{1}{x} \end{gathered}$

$\left(x_{k}\right)$ is Cauchy

f is continuous

$f\left(x_{k}\right)=k$ is not Cauchy

Given statement will be true if f is UNIFORMLY CONTINUOUS.