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$(X,\rho)$ metric space. If $S\subset X$ we define $\mathrm{dist}(x,S):=\mathrm{inf}\{\rho(x,y):y\in S\}$. Suppose $A\subset X$ sequentially compact. $(x_n)\subset X$ sequence such that lim dist$(x_n,A)=0$. $S:=\{x_n:n\in\mathbb{N}\}$. Could you help me to prove that $S\cup A$ is sequentially compact?

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