Is the inverse image of a Cauchy sequence of a continuous and injective function a Cauchy sequence?

Question

Let $X$ and $Y$ be two metric spaces

Let $f : X \to Y$ be a continuous and injective function

Let $\{x_n\}_{n \in \mathbb{N}} \subset X$ be a sequence in $X$ such that the sequence $\{f(x_n)\}_{n \in \mathbb{N}} \subset Y$ is a Cauchy sequence.

Is it true that $\{x_n\}_{n \in \mathbb{N}}$ is a Cauchy sequence too ?

Thanks.

Answer 1

Not in general. Let $X=Y=\mathbb R$. Assume that in $X$ you have the discrete metric and that in $Y$ you have the usual one. Let $f$ be the identity map and consider the sequence $\left(\frac1n\right)_{n\in\mathbb N}$. It is not a Cauchy sequence in $X$, but it is the preimage of $\left(\frac1n\right)_{n\in\mathbb N}$, which is a Cauchy sequence in $Y$.

Answer 2

Another example would be $X=Y=(0,\infty)$ both with the euclidean metric. Consider $ f(x) = 1/x$. Then the sequence $(1/n)_{n\geq 1}$ is Cauchy, but the preimage $(n)_{n\geq 1}$ is not.