Subsequence/Definition

Definition

Given a sequence $(x_n)_{n=1}^\infty$ we define a subsequence of $(x_n)^\infty_{n=1}$ as a sequence:

• $k:\mathbb{N}\rightarrow\mathbb{N}$ which operates on an $n\in\mathbb{N}$ with $n\mapsto k_n:=k(n)$ where:
  • $k_n$ is increasing, that means $k_n\le k_{n+1}$

We denote this:

• $(x_{k_n})_{n=1}^\infty$