$\le$ is a linear order on $\mathbb{N}$

Depends on

• Order on the natural numbers
• Addition of natural numbers
• Left identity for addition
• Addition is associative
• Addition is commutative
• Addition is cancellative
• Every nonzero natural number is a successor
• Trichotomy of the order on $\mathbb{N}$
• The von Neumann naturals form a Peano system

Used by

• ℝ((t⁻¹)) has the nested interval property for lengths tending to 0 Corollary
• Stolz-Cesaro, 0/0 form: if bₖ is strictly decreasing to 0, aₖ → 0, and the difference quotient converges, then aₖ/bₖ converges to the same value Corollary
• (ℕ, ≤) has no maximal element: Zorn's chain hypothesis fails Counterexample
• A free ultrafilter induces a finitely additive zero-one probability that is not countably additive Counterexample
• A progressive map with no fixed point, on a poset that is not chain-complete Counterexample
• A sequence with limsup = +∞: the greatest subsequential limit exists only in ℝ̄ Counterexample
• Null times divergent has no rule: xₖ = 1/k with yₖ = ck gives product limit c, and with yₖ = k² gives divergence Counterexample
• A summability (Toeplitz) matrix, the transformed sequence yₙ = ∑ₖ c_n,k xₖ, and regularity Definition
• Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences Definition
• The characteristic of a ring: the least n ≥ 1 with n · 1_R = 0 when one exists, and 0 otherwise Definition
• Well-order and well-ordered set Definition
• A positive sequence making all three inequalities of the ratio-to-root chain strict Example
• F^ℕ is a vector space and the eventually zero families form a linear subspace of it that is the span of the standard unit families Example
• min is a choice function on ℕ Example
• The 2-adic absolute value gives an ultrametric on ℚ, in which every triangle is isosceles and every point of a ball is a centre Example
• The block sequence 1/1; 1/2, 2/2; 1/3, 2/3, 3/3; … has subsequential limit set exactly [0,1] Example
• The standard unit families eₖ ∈ F^ℕ form a basis of the linear subspace of eventually zero families: an explicit infinite basis, built with no choice principle Example
• The subsets of ℕ containing a tail form the Fréchet filter, and it is proper and not an ultrafilter Example
• FALSE, once the ultrafilter lemma is available: every ultrafilter is principal False statement
• FALSE: every bounded sequence converges False statement
• FALSE: limits preserve strict inequalities False statement
• A Cauchy sequence with a convergent subsequence converges, to that subsequence’s limit Lemma
• A sequence has at most one limit Lemma
• A strictly increasing index map satisfies nₖ ≥ k Lemma
• An ordered field with the least-upper-bound property has the nested interval property and is Archimedean Lemma
• Cauchy completeness plus the Archimedean property imply the monotone convergence property Lemma
• Convergence depends only on the tail Lemma
• Every Cauchy sequence of reals is bounded Lemma
• Every convergent sequence is bounded Lemma
• Every convergent sequence is Cauchy Lemma
• Every real sequence has a monotone subsequence (the peak / rising-sun lemma) Lemma
• Every subset of an at most countable set is at most countable Lemma
• Finite sums re-indexed along an injection, with a zero term deleted, and concatenated; and the closure properties of linear independence: an independent list is injective and never 0_V, its sublists are independent, a list is independent exactly when it is injective with linearly independent image, and every subset of a linearly independent set is linearly independent Lemma
• For a prime p and a nonzero integer a: p^vₚ(a) ∣ a and p^vₚ(a)+1 ∤ a; pᵏ ∣ a holds exactly for k ≤ vₚ(a); vₚ(a) ≥ 1 exactly when p ∣ a; vₚ(1) = vₚ(-1) = 0; and vₚ(p) = 1 Lemma
• For every a > 0, a^1/n → 1 Lemma
• For every p > 0 and every positive rational α, n^α/(1+p)ⁿ → 0 Lemma
• For every real x, xᵏ/k! → 0 Lemma
• If xₖ ≤ yₖ eventually then limsup xₖ ≤ limsup yₖ and liminf xₖ ≤ liminf yₖ Lemma
• liminf xₖ ≤ limsup xₖ for every real sequence Lemma
• Limits preserve non-strict inequalities Lemma

…and 39 more results.

Dependency tree · two levels

16 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., §2.1-2.3 (Peano axioms, recursion, arithmetic) (standard reference, not scraped)
• Peano axioms (Wikipedia) (standard reference, not scraped)
• W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)