for asymptotic notation we need n which is $n\geq n_{0}$ , here $n_{0}$ is 10,000.So $g_{1}\left ( n \right )$ & $g_{2}\left ( n \right )$ will be compared after 10,000 range for n $\geq$ 10,000.
after 10,000 $g_{1}\left ( n \right ) = n^{2}$ & $g_{2}\left ( n \right ) = n^{3}$ i.e $g_{1}\left ( n \right )$ can never exceed $g_{2}\left ( n \right )$ in worst case.
So, $g_{1}\left ( n \right )$ $\leq$ C. $g_{2}\left ( n \right )$ $\Rightarrow$ $g_{1}\left ( n \right )$ $= $O$\left ( g_{2} \left ( n \right )\right )$ $\Rightarrow$ $g_{1}\left ( n \right ) = O\left ( n^{3} \right )$