Is a "bounded" and "Cauchy-continuous" function uniformly continuous? I have found lots of questions that ask whether "bounded" and "continuous" function is uniformly continuous. (I know the answer is no, by the way. e.g. $\sin(\frac 1 x)$ at $(0,1)$). However, if the image of a Cauchy sequence in $X$ is a Cauchy sequence in $Y$, and also $X$ bounded, is the function uniformly continuous?
Thank you.