How can I show that every orbit in a compact, non-empty, and invariant set $X$ is dense in $X$?

Question

First of all, $\phi_t(x)$ is the flow of a solution for the equation $x' = f(x)$, $f \in C^1$. To prove that every orbit in $X$ is dense in $X$, let $x \in X$. Since $X$ is invariant, we have $\phi_t(x)\in X, \forall t \ge0$. More than that, since $X$ is compact, there is a sequence of time $\{t_n\}_{n \in \mathbb{N}}$, and, $\{t_n\}\rightarrow \infty$ when $n \rightarrow \infty$, that $\phi_{t_n}(x) \rightarrow p$ when $n \rightarrow \infty$, for some $p \in X$. How can I go on?