Question

Give an example of a sequence that contains subsequences to every point in a sequence A = 1/n. Or justify why it cannot be possible.

Answer 1

what is the domain of n?

Answer 2

If \[n\in R \] then the sequence {1,...,2,...,3,...,\[\frac{1}{n}\]} for \[n\in[0,1]\] has every integer as a limit. e.g the subsequence {\[1\],\[\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},...\]} lets call it \[a_n\] converges to 1. Similarly it can also be shown that every point in the subsequence \[a_n\] also admits a subsequence. We could go on ad infinitum constructing subsequences, but we won't since we know that every cauchy sequence of real numbers converges, i.e real numbers cluster up together.