Limit superior and limit inferior

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

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

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 $(a_n{)}_{n=m}^{\mathrm{\infty }}$, denoted by $\underset{n\to \mathrm{\infty }}{lim sup}{a}_n$, by the formula:

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The mirror proceedure defines the limit inferior. First define:

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

The the limit inferior is:

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