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Proposition 3.4.3: Lim inf and Lim sup exist

lim sup and lim inf always exist (possibly infinite) for any sequence of real numbers.

Proof:

The sequence

Aj = inf{aj , aj + 1 , aj + 2 , ...}
is monotone increasing (which you should prove yourself). Hence, lim inf exists (possibly positive infinity).

The sequence

Bj = sup{aj , aj + 1 , aj + 2 , ...}
is monotone decreasing (which you should prove yourself). Hence, lim sup exists (possibly negative infinity).

Here we have to allow for a limit to be positive or negative infinity, which is different from saying that a limit does not exist.

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