Order on the reals

Depends on

• The real numbers
• Order on the rationals

Used by

• The Cauchy-sequence reals have the least-upper-bound property Corollary
• A null sequence times an unbounded sequence need not be null: (k+1)⁻¹·(k+1) = 1 Counterexample
• The quotient rule needs a nonzero limit in the denominator: 1/(k+1)⁻¹ = k+1 Counterexample
• The squeeze theorem needs the two bounds to share their limit Counterexample
• The sum of two divergent sequences can converge: k + (-k) Counterexample
• Circular phase distance modulo one Definition
• Contractive sequence: |xₖ₊₂ - xₖ₊₁| ≤ c |xₖ₊₁ - xₖ| for a fixed 0 < c < 1 Definition
• Divergence to +∞ and to -∞ Definition
• Intervals of ℝ: the nine order-convex forms, nondegeneracy, and length Definition
• Limits and Cauchy sequences of reals Definition
• Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric Definition
• Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of ℝ, with the dictionary to monotone sequences Definition
• Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences Definition
• Open ball, closed ball and sphere in a metric space Definition
• Open subset of ℝ (every point has a neighbourhood inside it), closed subset (complement open), and clopen Definition
• Sequences of reals: bounded, eventually, frequently, tails, subsequences Definition
• The closed long ray ω₁ × [0,1) under the lexicographic order, and the long line, with the order topology Definition
• The extended real line ℝ̄ = ℝ ∪ {-∞, +∞}, its order, and the arithmetic that is left undefined Definition
• The order topology of a linearly ordered set, with the open rays as a subbasis; order-convex sets, order-density, the least upper bound property, and linear continua Definition
• The ε-neighbourhood and the punctured ε-neighbourhood of a point of ℝ Definition
• The ε-δ limit lim_x → c f(x) = L of f : A → ℝ at a limit point c of A Definition
• Assuming the Axiom of Choice, compactness of [0,1] derived from the subbase lemma alone, using only the rays as a subbasis and the least upper bound property Example
• Følner sets in ℝⁿ Example
• Infinite Ramsey for pairs gives a nondecreasing or nonincreasing subsequence of every real sequence Example
• ℝ ≈ P(ℕ) in ZF, by the Cantor set for one injection and by the cuts {q ∈ ℚ : q < x} for the other; so | ℝ | = 2^ℵ₀ under the Axiom of Choice Example
• ℝ with the half-open intervals [a,b) as a basis is not compact and, assuming the Axiom of Countable Choice, is Lindel"of, while its square is not Lindel"of, the antidiagonal being an uncountable closed discrete subspace Example
• The absolute value of the limit, and the limit of the absolute value Example
• The alternating sequence (-1)ᵏ/(k+1) converges to 0 Example
• The order topology on a totally ordered set, with the open rays as a subbasis, and its agreement with the usual topology of ℝ Example
• The sequence 1/(k+1) converges to 0 Example
• The square roots √k diverge to +∞ Example
• FALSE: every bounded sequence converges False statement
• FALSE: limits preserve strict inequalities False statement
• Every real sequence has a monotone subsequence (the peak / rising-sun lemma) Lemma
• Every subset of ℝ̄ has a least upper bound and a greatest lower bound in ℝ̄, agreeing with the real supremum and infimum on nonempty sets bounded in ℝ Lemma
• For positive terms, null and divergence to +∞ are reciprocal Lemma
• Measurable dense selections for fields of nonempty compact sets Lemma
• Measurable Gram-Schmidt and constant-field trivializations on dimension strata Lemma
• Nonnegativity of a metric is a consequence of the other axioms, not an axiom Lemma
• The absolute value is compatible with limits Lemma

…and 8 more results.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• T. Tao, Analysis I, 3rd ed., §5.4 (standard reference, not scraped)
• L. S. Krapp, Constructions of the real numbers: a set theoretical approach (Oxford, 2014) (standard reference, not scraped)