Let $$(a_n)_{n=0}^{\infty}$$ and $$(b_n)_{n=0}^{\infty}$$ be sequences of reals. We say that $$(b_n)_{n=m}^{\infty}$$ is a subsequence of $$(a_n)_{n=m}^{\infty}$$ if there exists a function $$f: \mathbb{N} \rightarrow \mathbb{N}$$ which is [...] such that
$b_n = a_{f(n)} \text{ for all } n \in \mathbb{N}$
Let $$(a_n)_{n=0}^{\infty}$$ be the sequence represented by the function $$g: \mathbb{N} \rightarrow \mathbb{R}$$. If $$f: \mathbb{N} \rightarrow \mathbb{N}$$ is [...], then the sequence defined by [...] is a subsequence of $$(a_n)_{n=0}^{\infty}$$.