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**zzzhhh** Let $\displaystyle x$ be a limit point of $\displaystyle A'$, $\displaystyle U$ an arbitrary neighborhood of $\displaystyle x$. To prove that $\displaystyle x\in A'$ so that $\displaystyle A'$ is closed, we must find an element $\displaystyle y$ in $\displaystyle U$ satisfying $\displaystyle y\in A$ and $\displaystyle y\ne x$. Since $\displaystyle x$ is a limit point of $\displaystyle A'$, there is in $\displaystyle U$ a point $\displaystyle x'\in A'$ and $\displaystyle x'\ne x$.