Question

Is every subsequence of a divergent sequence is divergent ? Justify.

Answer 1

Let us consider two sequences:

$\left(a_k\right)_{k=1}^{\infty}, a_k = \dfrac{1}{k}; \\ \left(b_k\right)_{k=1}^{\infty}, b_k = (-1)^k k.$

We construct the sequence $(c_k)_{k=1}^{\infty}$ in form of $a_1, b_1, a_2, b_2, a_3, b_3,$ and so on.

This sequence is divergent, because we can consider the subsequences with different limits: $c_2 = -1, c_6 = -3, c_{10} = -5, \ldots , c_{4k+2} = -(2k+1), \ldots$ , which has the limit of $-\infty$

and the subsequence $c_4 = 2, c_8=4, \ldots, c_{4k}=2k, \ldots,$ which has the limit of $+\infty.$

But subsequence $c_1, c_3, \ldots, c_{2k-1} = \dfrac{1}{k},\ldots,$ has the limit of 0, so it is not divergent.