Completeness of the real numbers
Objective. Use bounds, suprema and completeness.
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- 2. Worked example
- 3. Practice
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- 5. Continue
Learn: the key idea
Completeness of the real numbers is the property that every non-empty set bounded above has a least upper bound (supremum) lying in ℝ itself, unlike in ℚ. The supremum s = sup(S) satisfies s ≥ x for all x ∈ S, and for any ε > 0 there is some x ∈ S with x > s − ε. Infima are defined dually for sets bounded below. For example, S = {x ∈ ℚ : x² < 2} is bounded above in ℚ but has no rational supremum, whereas in ℝ, sup(S) = √2. Completeness underlies convergence of Cauchy sequences and the existence of limits used throughout analysis.
Worked example
Find sup(S) and inf(S) where S = {1 − 1/n : n ∈ ℕ, n ≥ 1}, justifying completeness is needed.
- 1Elements are 0, 1/2, 2/3, 3/4, … approaching but never reaching 1.
- 2Every element satisfies 1 − 1/n < 1, so 1 is an upper bound.
- 3For any ε > 0 choose n > 1/ε so that 1 − 1/n > 1 − ε, so no smaller upper bound works.
- 4The smallest element is n = 1 giving 0, so inf(S) = 0 and sup(S) = 1.
Answer: sup(S) = 1, inf(S) = 0
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Start practisingCommon mistake. Students assume a bounded set always has a supremum that belongs to the set; sup({x : x < 3}) = 3, but 3 ∉ the set. Always check whether the bound is attained before calling it a maximum.
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