## Proposition 3.4.3: Lim inf and Lim sup exist

### Proof:

The sequence

is monotone increasing (which you should prove yourself). Hence,A_{j}= inf{a_{j}, a_{j + 1}, a_{j + 2}, ...}

*lim inf*exists (possibly positive infinity).

The sequence

is monotone decreasing (which you should prove yourself). Hence,B_{j}= sup{a_{j}, a_{j + 1}, a_{j + 2}, ...}

*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.