Squaring is monotone on the nonnegatives

Depends on

• Ordered field
• Sign rules for products and monotonicity of multiplication
• Multiplication by zero: $0 \cdot a = 0$
• Basic properties of the absolute value

Used by

• The induced length is a norm Corollary
• {q ∈ ℚ : q ≥ 0, q² < 2} is closed and bounded in ℚ and is not compact Counterexample
• f(t) = (t², t³) on [0,1]: no ξ satisfies f(1)-f(0) = f'(ξ) Counterexample
• ℚ ∩ [0,2] is bounded and disconnected, so being an interval of ℚ is not enough Counterexample
• x ↦ √x is a uniformly continuous bijection of [0,∞) onto itself whose inverse x ↦ x² is not uniformly continuous Counterexample
• √· on [0,∞) is uniformly continuous and exactly 1/2-Hölder, and is not Lipschitz Example
• √x is absolutely continuous but not Lipschitz on [0,1] Example
• A convergent sequence in ℝ³ and the integral ∫₀¹ (1, t, t²), computed componentwise Example
• Hilbert spaces are uniformly convex Example
• Steinitz's confinement bound realised on an explicit list of six unit vectors in ℝ² summing to zero Example
• sup{q ∈ ℚ : q > 0, q² < 2} = √2 in ℝ, and no supremum in ℚ Example
• The comparison constants between ‖·‖₁, ‖·‖₂ and ‖·‖_∞ on ℝ², and vectors attaining each Example
• The cube [-M,M]ⁿ in ℝⁿ is totally bounded, with an explicit finite ε-net of grid points and no appeal to the integer part Example
• The map x ↦ (x + 2/x)/2 is a contraction of [1,2] with fixed point √2, and the a priori bound gives the error after n steps Example
• The metrics d₁, d₂ and d_∞ on ℝⁿ are metrics and are Lipschitz equivalent, with explicit constants Example
• FALSE: in every ordered field a closed bounded set is compact, so Heine-Borel needs no completeness False statement
• Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive Lemma
• Equality in the unit-complex finite-sum bound Lemma
• For n ≥ 1 the product topology on n copies of the usual topology of ℝ is the metric topology of d_∞ on ℝⁿ, and hence also of d₁ and d₂, so ℝⁿ as a product and ℝⁿ as a metric space are one space Lemma
• ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it Lemma
• Square-summable orthogonal families have norm-convergent finite sums Lemma
• The finite and reverse triangle inequalities for a norm; and for n ≥ 1 every norm N on ℝⁿ satisfies N(x) ≤ C‖ x‖₁ and is Lipschitz, hence continuous, for d₂ Lemma
• Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation Theorem
• Cauchy–Schwarz: |⟨ x,y⟩|≤‖x‖ ‖y‖, with equality exactly for dependent pairs Theorem
• Every complex number has a square root, by an explicit Cartesian formula Theorem
• For a ≤ b and f : [a,b] → ℝᵐ integrable when a<b, ‖∫ₐᵇ f‖₂ ≤ ∫ₐᵇ ‖ f‖₂; for a<b, ‖ f‖₂ is integrable Theorem
• Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line Theorem
• Jordan–von Neumann: a norm is induced by an inner product exactly when it satisfies the parallelogram law Theorem
• ℝ and ℝⁿ for n ≥ 1 with the Euclidean metric are complete, componentwise from the Cauchy criterion in ℝ Theorem
• Square roots exist: a unique √a ≥ 0 with (√a)² = a; the positives are {x² : x ≠ 0} Theorem
• The Cauchy-Schwarz inequality for finite sums Theorem

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

• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• M. Spivak, Calculus, 4th ed., Ch. 1 (standard reference, not scraped)