Lim inf and Lim sup exist

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

The sequence

A_j = inf{a_j , a_{j + 1} , a_{j + 2} , ...}

is monotone increasing (which you should prove yourself). Hence, lim inf exists (possibly positive infinity).

The sequence

B_j = sup{a_j , a_{j + 1} , a_{j + 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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