• 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(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_j and B_j mentioned in the definition.

Let's find the numbers A_j = inf{a_j, a_{j + 1}, a_{j + 2}, ...} for the sequence {-1, 1, -1, 1, ...}.

• A₁ = inf{-1, 1, -1, 1, ...} = -1
• A₂= inf{1, -1, 1, -1, ... } = -1

and so on. Therefore, it is clear that

lim inf = -1

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

• B₁ = sup{-1, 1, -1, 1, ...} = 1
• B₂ = sup{1, -1, 1, -1, ...} = 1

and so on. Therefore, it is clear that

lim sup= 1

Interactive Real Analysis, ver. 2.0.1(c)
(c) 1994-2018, Bert G. Wachsmuth
Page last modified: Jan 26, 2023