Sequences

Sequence

A sequence is a function is any function

f

such that

dom (f) = I

where

I

either finite or countable.

With this in place, we can consider the $n$ -th element by $f (n)$ . Since we can think of the sequence as being enumerated, the notation $a_{n}$ is employed for $f (n)$ where $a$ is usually any choice of lower case alphabet characters like $a, b, c, d, \dots$ . That said, we can notate the entire sequence by $(a_{n})$

Subsequence

Given a sequence

(a_{n}) : N_{1} \to R

then given any strictly increasing map

σ : N_{1} \to N_{1}

we say that the sequence

x_{n} := (a_{σ (n)})

is a subsequence, and we usually denote it just as

a_{σ (n)}