Given the ODE system $x'=y, y'=-h(y)g(x)$ where $h,g:\mathbb{R}\to\mathbb{R}$ continuous and $h(y)>0$ and $x.g(x)>0$, for all $x\neq0$ Determine sufficient condition on the existence of the solution of the ODE. Consider $V(x,y)=\int^y_0\frac{s}{h(s)}ds+\int^x_0g(s)ds$ the Lyapunov function.
My attempt: I've shown that $V'=0$, however, I don't know how to show the uniform convergence to infinity.