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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:

[...]

The mirror proceedure defines the limit inferior. First define:

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

The the limit inferior is:

[...]