Math and science::Analysis::Tao::06. Limits of sequences
Suprema and infima of sequences, definition
Carrying over the idea of supremum and infimum of sets of reals to supremum and inimum of sequences of reals.Let \( (a_n)_{n=m}^{\infty} \) be a sequence of real numbers. Then we define \( \sup(a_n)_{n=m}^{\infty} \) to be the supremum of the set \( \{a_n : n \ge m\} \), and \( \inf(a_n)_{n=m}^{\infty} \) to be the infimum of the same set \( \{a_n : n \ge m\} \).
The quantities \( \sup(a_n)_{n=m}^{\infty} \) and \( \inf(a_n)_{n=m}^{\infty} \) are sometimes written as \( \sup_{n \ge m}a_n \) and \( \inf_{n \ge m}a_n \) respectively.
Sequences as functions
When sequences are viewed as functions from \( \mathbb{N} \) to \( \mathbb{R} \), the supremum of the sequence can be viewed as the supremum of the codomain of the function.