Sequence

A sequence is a function is any function \( f \) such that \( \text{dom} \left ( f \right ) = I \) where \( I \) either finite or countable.

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

Subsequence

Given a sequence \( \left( a _ n \right) : \mathbb{ N } _ 1 \to \mathbb{ R } \) then given any strictly increasing map \( \sigma : N _ 1 \to N _ 1 \) we say that \( \left( a _ \sigma \left( n \right) \right) \) is a subsequence.