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Math and science::Analysis::Tao::06. Limits of sequences

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.