Assume a dynamical system $\dot{x}=f(x)$, $x \in R^n$ (with $f$ sufficiently smooth -- see below) satisfies the following:

The box $B=[-1,1]^n$ is forward invariant: any trajectory that starts in $B$ stays in $B$;

The system has a finite number of equilibrium points in $B$;

There is a smooth function $V(x) \ge 0$ such that $\dot{V}(x) < 0$ for all points in $B$ except the fixed points. In particular, the only periodic trajectories are the trivial ones corresponding to the fixed points.

I would like to conclude that any trajectory starting in $B$ must approach in the limit one of the fixed points.

Is this actually correct? What hypotheses on $f$ are necessary/sufficient?

This looks like a basic question, so I would be happy with just a reference to a standard text.