• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 3.1. Sequences
• 3.2. Cauchy Sequences
• 3.3. Subsequences
• 3.4. Lim Sup and Lim Inf
• 3.5. Special Sequences
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

3.4. Lim Sup and Lim Inf

Examples 3.4.2(b):

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

Interactive Real Analysis, ver. 2.0.1(c)
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Page last modified: Apr 25, 2024