In one paper the author uses the statement without citation:

Let $(M,g)$ be a Riemannian manifold. The gradient $\nabla F$ of a proper function $F: M\rightarrow \mathbb{R}$ is integrable vector field, i.e. its integral curves are defined for all times.

In this particular case, $F$ is also bounded from below, and the Riemannian metric $g$ on $M$ is complete.

My question is

1)Is this statement true in general?

2)If not, why is the statement true when additional assumptions are satisfied (function is bounded from below and metric is complete)?

I could not find a reference on this in the literature; if someone knows it - that will be enough.

Thanks for any help!