Suppose $f$ is a uniformly continuous mapping from a metric space $X$ to a metric space $Y$. Prove that $f\left(x_{n}\right)$ is a Cauchy sequence in $Y$ for every Cauchy sequence $x_{n}$ in $X$.

Let $f$ be a continuous mapping between metric spaces and suppose that $f$ has the property that $f\left(x_{n}\right)$ is a Cauchy sequence whenever $x_{n}$ is a Cauchy sequence. Is it true that $f$ must be uniformly continuous? Justify your answer.