Consider the system of differential equations $\dot{x}(t) = f(t,x)$ where $x \in \mathbb{R}^n$ and $f$ is a $C^{\infty}$ function. Suppose that every trajectory which begins in the closed unit ball tends to zero as $t \rightarrow \infty$. Does it follow that there exists a ball around the origin $\mathcal{B}$ such that every trajectory that begins in the unit ball stays in $\mathcal{B}$?
My natural inclination is to think the answer ought to be yes and one should be able to show this by picking a convergent subsequence somehow. I'm having trouble making this work, however.
Here is where I am stuck. Supposing $x_i(t)$ is a trajectory that begins from $x_i(0)$ at $t=0$ and has distance at least $i$ away from the origin at some later time; and supposing $\lim_{i \rightarrow \infty} x_i(0) = x$; then I can't see how to derive a contradiction between the fact that the trajectory beginning from $x$ approaches zero, and is consequently bounded, and the the fact that $||x_i(t_i)||_2 \geq i$ for some $t_i$. If each $t_i$ was below some $T$ a contradiction is easy to obtain, but how to deal with $t_i$'s which blow up?
Finally, this is not homework.