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

# Limit superior and limit inferior

Suppose that $$(a_n)_{n=m}^{\infty}$$ is a sequence. We define a new sequence $$(a_N^+)_{N=m}^{\infty}$$ by the formula:

$a_N^+ := \sup(a_n)_{n=N}^{\infty}$

In other words, $$a_N^+$$ is the supremum of all the elements in the sequence from $$a_N$$ onwards.

We then define the limit superior of the sequence $$(a_n)_{n=m}^{\infty}$$, denoted by $$\limsup_{n\rightarrow \infty}a_n$$, by the formula:

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The mirror proceedure defines the limit inferior. First define:

$a_N^- := \inf(a_n)_{n=N}^{\infty}$

The the limit inferior is:

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