# Is a sequence of limits bounded

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).

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.