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 \geq m}

, and

inf (a_{n})_{n = m}^{\infty}

to be the infimum of the same set

{a_{n} : n \geq m}

The quantities $sup (a_{n})_{n = m}^{\infty}$ and $inf (a_{n})_{n = m}^{\infty}$ are sometimes written as $sup_{n \geq m} a_{n}$ and $inf_{n \geq m} a_{n}$ respectively.

Sequences as functions

When sequences are viewed as functions from $N$ to $R$ , the supremum of the sequence can be viewed as the supremum of the codomain of the function.

03.11.2019