$X$ is a $T_2$ space. Let $fin C_0(X), gin C_c(X)$. Prove that, $fgin C_c(X)$.

Here $C_0(X)$ denotes the collection of all such continuous maps $f:XtoBbb{C}$ such that $forall epsilon>0 exists $ compact set $K$ such that $|f(x)|<epsilon forall xin K^c$.

And $C_c(X)$ denotes the collection of all such continuous maps $g:XtoBbb{C}$ such that $text{supp}(g):=overline{{xin X|g(x)ne 0}}$ is compact.

It’s easy to see that $C_c(X)subset C_0(X)$.

Suppose $fin C_c(X)$. Then $text{supp}(fg)subset text{supp}(f)cuptext{supp}(g)$ and since union of two compact sets is compact and $text{supp}(fg)$ is closed, $text{supp}(fg)$ is compact, hence $fgin C_c(X)$.

But I’m stuck with the case $fin C_0(X)setminus C_c(X)$?

Can anyone help me in this regard? Thanks for your help in advance.