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**darkchibi07** Let $\displaystyle A \doteq \{x_n\}^\infty_{n \doteq 1}$ be a sequence and let $\displaystyle B$ be the set of all subsequential limits of $\displaystyle A$. I have to show that $\displaystyle A \cup B$ is closed.

I got that $\displaystyle B$ is closed from that theorem in Rudin's textbook that said that the subsequential limits form a closed set which leads that $\displaystyle B$ is closed. Now I figured that I should prove that $\displaystyle A$ is closed as well in order to complete the proof, and that's where I'm stuck at.