equilibrium of a flow

Flow $\phi : \mathbb R \times X \rightarrow X$ on metric space $(X, d)$, we know there is a sequence of points $x_n \in X$ s.t. the orbit through $x_n$ is a periodic orbit with period $T_n > 0$, also $\lim_{n \to \infty} x_n = x_*, \lim_{n \to \infty} T_n = 0$ for some $x_* \in X$. How to prove that $x_*$ is an equilibrium of the flow?

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I assume that $\phi$ is continuous on $\mathbb R\times X$. Given $t>0$, you will be able to find an integer sequence $(k_n)$ such that $k_nT_n\to t$. Then $\phi(k_nT_n,x_n)\to \phi(t,x_*)$ by continuity, but at the same time $\phi(k_nT_n,x_n)=x_n\to x_*$.