Proving that the solutions of a system of ODE $\frac{dx}{dt}=-\nabla V(x)$ are defined for all positive time. Let $V:\mathbb{R}^n \to \mathbb{R}$ s.t $V\in C^2$ and $\lim_{x\to \infty} V(x)=\infty$. Prove that the system: 
$$\frac{dx}{dt}=-\nabla V(x)$$ 
has defined solutions for all positive time.
My questions: is the hypothesis right? How can I use the limit?
Also, thanks for any suggestions.
 A: Multiply (dot-product) with $\frac{dx}{dt}$ to find
$$\frac{dx}{dt}\cdot \frac{dx}{dt} = -\frac{dx}{dt}\cdot\nabla V \implies \left(\frac{dx}{dt}\right)^2 = -\frac{d V}{dt}$$
where we have used $\frac{dV}{dt} = \frac{\partial V}{\partial t} + \sum_{i=1}^3\frac{\partial V}{\partial x_i}\frac{dx_i}{dt} = \nabla V \cdot \frac{dx}{dt}$ and the fact that $V$ does not depend explicitly on $t$ so $\frac{\partial V}{\partial t} = 0$. By integrating up we find
$$V(x(0)) - V(x(t)) = \int_0^t\left(\frac{dx}{dt}\right)^2dt$$
The term on the right hand side is a stricktly increasing (and positive for $t\geq 0$) function of $t$ since $\left(\frac{dx}{dt}\right)^2 \geq 0$. 
Now lets assume that $x(t)$ does not stay bounded, i.e. we assume that there exist a $t_*$ s.t. $\lim_{t\to t_*}|x(t)| = \infty$. We then have
$$V(x(0)) - V(\infty) = \int_0^{t_*}\left(\frac{dx}{dt}\right)^2dt \geq 0$$
which contradicts $ V(\infty) = \infty$. We can therefore conclude that $x(t)$ stays bounded at all times, i.e. that the solutions of the ODE are defined at all times.
