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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 $\phi(k_nT_n,x_n)=x_n\to x_*$.

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