If $X$ is a gradient-like vector field of a Morse function $f\colon M\to \mathbb{R}$, then the integral curve $c_p(t)$ starting at an arbitrary point $p$ approaches critical points as $t\to \pm \infty$.
This is a paragraph in Matsumoto's Morse theory and I'm having a bit of trouble understanding it. I assume that $M$ should be compact in this context, otherwise $f$ may not have any critical points or a global integral curve.
Can someone give me a hint on why this should be true? (I wonder if it is considered obvious by the author, or as a fact that he expects from the reader to take on faith.) Thanks!