Let's assume $f: \mathbb{R}^d \to \mathbb{R}^d$ is continuous and locally lipschitz. Consider the autonomous ODE $x' = f(x)$. Assume $\exists R > 0. \forall n \in \mathbb{N}. \exists \varphi_n: \mathbb{R} \to \mathbb{R}^d$ which is a $\frac 1 n$-periodic solution and such that $\forall t \in \mathbb{R}. \varphi_n(t) \in \stackrel{-}{B}(0,R)$. Proof that $\exists p \in \stackrel{-}{B}(0,R). f(p) = 0$.
I have the intuition that maybe I could work in $(C[0,1],\|\cdot\|_{\infty})$ extract a converging partial $\varphi_n \to \varphi$ and prove that the limit has to be $0$-periodic. In the language of dynamical systems, we have a sequence of cycles with radius converging to $0$ and have to show that they have a equilibrium as a limit point. What is a nice way of proving this?
Edit
I think that using a Poincaré map can be of use here.