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If you take an infinite sequence of infinite sequences and then find that the first element in every sequence converges to a point (because it is Cauchy and in a complete space), and then the second element in every sequence converges...and so on. If you take the sequence of those limit points, is it bounded?

EX) $\{a_1, a_2, a_3,....\};\\ \{b_1, b_2, b_3,....\};\\ \{c_1, c_2, c_3,....\};\cdots$

Where $\{a_1, b_1, c_1,....\}$ is Cauchy and converges to a point, and $\{a_2, b_2, c_2,....\}$ is Cauchy and converges to a point, etc. Then how would I show that the sequence of the limit points is bounded. (Knowing that the sequence of sequences is Cauchy and bounded and the each column is also Cauchy and bounded).

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Suppose that the sequence of $n$-terms is the constant $n$ $a_1=1,b_=1,c_1=1; a_2=2,b_2=2,c_2=2,...$, each of these sequences converges converges but their limit points is not bounded.

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