Real Analysis True or False Any subsequence of a subsequence

Real Analysis

True or False: Any subsequence of a subsequence of a monotonic sequence has a limit which is defined.

Solution

a sequence (x_n), a subsequence, notated as (x_nj)_j=1 , is a sequence where (n_j) is strictly increasing sequence of natural numbers. Phrased differently, a subsequence is a sequence where (n_j) is an interval between the subscriped number and the superscripted number, from the notation.

For example, taking n_j=2j would the subsequence consisting of every other element of the original sequence, that is (x₂, x₄, x₆, …)