Example: What is inf, sup, lim inf and lim sup for ?

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Since this sequence is {1, 1/2, 1/3, 1/4, ...} the infimum is zero, while the supremum is 1. As for lim inf and lim sup, we find first the sequence of numbers A_j and B_j mentioned in the definition.

• A₁ = inf{1, 1/2, 1/3, 1/4, ...} = 0
• A₂ = inf{1/2, 1/3, 1/4, 1/5, ...} = 0
• A₃ = inf(1/3, 1/4, 1/5, 1/6, ...} = 0

and so on. Therefore, it is clear that

• lim inf = 0

Similarly, we find the numbers B_j= sup{a_j, a_{j + 1}, a_{j + 2}, ...}:

• B₁ = sup{1, 1/2, 1/3, 1/4, ...} = 1
• B₂ = sup{1/2, 1/3, 1/4, 1/5, ...} = 1/2
• B₃ = sup(1/3, 1/4, 1/5, 1/6, ...} = 1/3

and so on. Therefore, it is clear that

• lim sup = 0

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