Equicontinuity and uniform convergence 2

Let $\{f_n\}_n$ be a sequence of real valued functions on a compact metric space $K$. Suppose that for all $x$ we have $f_n(x) \to f(x)$ as $n \to \infty$ and that the family $\{f_n\}_n$ is equicontinuous. Prove that $f_n \to f$ uniformly.

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  1. Using the assumption of equi-continuity, one can show that $f$ is necessarily continuous. So we can assume $f=0$.

  2. Assume by contradiction that $(f_n)_n$ doesn’t converge to $0$ uniformly. Then, after having taken a subsequence, there is $\delta$ such that $\lVert f_{k’}\rVert >\delta$ for some $\delta$ valid for all $k’$. By compactness and continuity, $\lVert f_{k’}\rVert=|f_{k’}(x_{k’})|$ for some $x_{k’}$.

  3. Extract from $(x_{k’},k’\in\Bbb N)$ a convergent subsequence, and use equi-continuity to reach a contradiction.