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I am having trouble understanding the answer in this question: Is this proof that $c_0$ is a closed subspace of $\ell^\infty$ correct? Also need some help finish it.

In the first answer by Alonso Delfín, what does the n in closed bracket as superscript mean? I know that sequences are denoted as as x subscript n. Finally, in the fourth line of the answer, it is shown to have both, superscript, (n) and subscript, j on x. If someone can clarify what these mean with an example, that would be greatly appreciated.

Thank you in advance.

Sean
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  • It is written that $(x^k)$ is a sequence of sequences. This means that every $x^k$ is in turn a sequence: $x^K= (a_1^k, a_2^k, \ldots)$ where each $a_i^k$ is a number. – Mauro ALLEGRANZA Sep 28 '21 at 08:55
  • Thank you but I am talking about the one in the first answer, where we have x superscript (n) and then also with j as a subscript. I am getting confused. An example would be very helpful. @MauroALLEGRANZA – Sean Sep 28 '21 at 08:57

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