Let $f$ be a uniformly continuous function from a subset $E$ of a metric space $X$ to metric space $Y$ and $E$ is bounded in $X$. Is $f(E)$ a bounded subset of $Y$? What if $Y$ is complete metric space?
Well, I have this thought: Suppose $f:\Bbb R_{()}\to \Bbb R$ defined by $f(x)=x$ where $\Bbb R_{()}$ has bounded metric. Now let $E=\Bbb R$, then $f(E)$ not bounded even if $E$ bounded and $f$ uniformly continuous.
Am I right?
It seems to me that you are right.