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It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff

$$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X \quad x_{n} \rightarrow p \Rightarrow f(x_{n}) \rightarrow f(p)$$

It is also well known that if $X$ and $Y$ are metric spaces and $f : X \rightarrow Y$ is uniformly continuous, then $f$ maps Cauchy sequences to Cauchy sequences.

By analogy it seems plausible that if a function between metric spaces maps Cauchy sequences to Cauchy sequences then it must be uniformly continuous. However mimicking the proof of the analogous result for continuous maps doesn't work, which makes me think the result if false. Does anyone know any counterexamples?

Also on the uniform continuity wikipedia page, it says that the result is true if $X$ and $Y$ are subsets of $\mathbb{R}^{n}$. EDIT: It actually doesn't say this, I misread the page.

It is well known that if $X$ is a first countable topological space and $Y$ is a topological space, then $f : X \rightarrow Y$ is continuous iff
$$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X \quad x_{n} \rightarrow p \Rightarrow f(x_{n}) \rightarrow f(p)$$
It is also well known that if $X$ and $Y$ are metric spaces and $f : X \rightarrow Y$ is uniformly continuiouscontinuous, then $f$ maps Cauchy sequences to Cauchy sequences.
Also on the wikipedia page, it says that the result is true if X $X$ and Y $Y$ are subsets of $\mathbb{R}^{n}$\mathbb{R}^{n}$. 2 edited body; added 1 characters in body; deleted 5 characters in body; deleted 2 characters in body It is well known that if$X$is a first countable topological space and$Y$is a topological space, then$f : X \rightarrow Y$is continuous iff $$\forall x \in {\rm map}(\mathbb{N},X),\forall p \in X \quad x_{n} \rightarrow p \Rightarrow f(x_{n}) \rightarrow f(p)$$ It is also well known that if$X$and$Y$are metric spaces and$f : X \rightarrow Y$is uniformly continuious, then$f$maps Cauchy sequences to Cauchy sequences. By Analogy it seems plausible that if a function between metric spaces maps cauchy sequences to cauchy sequences then it must be uniformly continuious, however . However mimicking the proof of the analogus result for continuious maps doesn't work, which makes me think the result if false. Does anyone know any counter examples? Also on the wikipedia page, it says that the result is true if X and Y are finite dimensional Banach spaces.subsets of$\mathbb{R}^{n}\$