3.4. Lim Sup and Lim Inf
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 Aj and Bj mentioned in the definition.Let's find the numbers Aj = inf{aj, aj + 1, aj + 2, ...} for the sequence {-1, 1, -1, 1, ...}.
- A1 = inf{-1, 1, -1, 1, ...} = -1
- A2= inf{1, -1, 1, -1, ... } = -1
lim inf = -1Similarly, we find the numbers Bj = sup{aj, aj + 1, Aj + 2, ...}:
- B1 = sup{-1, 1, -1, 1, ...} = 1
- B2 = sup{1, -1, 1, -1, ...} = 1
lim sup= 1