let $(X,d)$ be a COMPACT metric space. and $f$ is continuous function on $X$ that maps $X$ to real numbers, prove $f^2$ (pointwise product) is uniformly continuous.
I know by theorem, that $f$ is uniformly continuous on $X$.
I know is general this is not true that : if $f$ is uniformly continuous, then $f.f=f^2$ is uniformly continuous.
But can I say since $X$ is compact, so is $X\times X$. Hence $f^2$ is uniformly continuous on $X\times X$. ( and we know if $f$ is continuous on $X$, so is $f^2$ on $X\times X$).
Can this be considered as a proof for this question?