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