Let $(X, d)$ a metric space, $F: X \rightarrow \mathbb{R}$ a continuous and bounded function, and for all $n \in \mathbb{N}$ $\alpha_n: X \rightarrow X$ a function such that $(\sup_{x \in X} d(\alpha_n(x), x): n \in \mathbb{N})$ converges to zero. Let $F_n = F \circ \alpha_n$. Show that $F_n$ is equicontinuous and converges uniformly.
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