## Examples 3.4.2(a):

Clearly, the infimum of the sequence is*-1*, and the supremum is

*+1*. To find

*lim inf*and

*lim sup*, we will first find the sequence of numbers

*A*and

_{j}*B*mentioned in the definition.

_{j}
Let's find the numbers
*A _{j} = inf{a_{j}, a_{j + 1}, a_{j + 2}, ...}*
for the sequence

*{-1, 1, -1, 1, ...}*.

*A*_{1}= inf{-1, 1, -1, 1, ...} = -1*A*_{2}= inf{1, -1, 1, -1, ... } = -1

Similarly, we find the numberslim inf = -1

*B*:

_{j}= sup{a_{j}, a_{j + 1}, A_{j + 2}, ...}*B*_{1}= sup{-1, 1, -1, 1, ...} = 1*B*_{2}= sup{1, -1, 1, -1, ...} = 1

lim sup= 1