$(8$ points$)$ Let $f$:$R^m{R}^n$ and $g$:$R^n{R}^{p}be$ functions such that

$\underset{x0}{lim}\frac{|f(x)|}{|x|}=0,$ and $,\underset{y0}{lim}\frac{|g(y)|}{|y|}=0.$

Prove directly, using the definition and properties $oflimits,$ that

$\underset{x0}{lim}\frac{|g(f(x))|}{|x|}=0$