The triangle inequality

Depends on

• Basic properties of the absolute value
• Order is preserved by adding a constant and by adding inequalities
• Ordered field

Used by

• The reverse triangle inequality Corollary
• The total-variation bound for a Riemann–Stieltjes integral Corollary
• The uniform limit of uniformly continuous real-valued functions is uniformly continuous Corollary
• Kolmogorov extension alone does not give a continuous version Counterexample
• On a closed interval of ℚ there is a continuous unbounded function, a bounded one with no maximum, and one without the intermediate value property Counterexample
• The hyperbola {(x,y) : xy = 1} is closed in ℝ² and its image under the first projection is ℝ ∖ {0}, which is not closed Counterexample
• Two copies of ℝ glued along ℝ ∖ {0} give a non-Hausdorff quotient of a metrizable space, by an open quotient map Counterexample
• ℤ is closed and not compact, and (0,1) is bounded and not compact: neither hypothesis of Heine-Borel can be dropped Counterexample
• A norm on a real vector space, the induced metric, and the dictionary with the metric axioms Definition
• Limits at +∞ and -∞, and infinite limits at a point Definition
• Sequences, convergence, Cauchyness, monotonicity, boundedness and closed intervals in an arbitrary ordered field Definition
• The oscillation ω_f(S) = sup{ |f(x) - f(y)| : x, y ∈ S } of f on a set and the oscillation ω_f(c) = inf_δ > 0 ω_f(A ∩ N_δ(c)) at a point, both taken in the extended reals Definition
• The topology of compact convergence on C(X,Y) for metric X and Y: uniform convergence on each compact subset of X Definition
• The ε-neighbourhood and the punctured ε-neighbourhood of a point of ℝ Definition
• The absolute-value function is convex Example
• The diagonal of ℝ is closed in ℝ², computed from the product basis Example
• The Hilbert cube [0,1]^ℕ with the product topology is metrizable, by d(x,y) = ∑ₖ |xₖ - yₖ| / 2^ k+1 Example
• The map (x,z) ↦ x · z on ℝ × ℝ and its transpose z ↦ (x ↦ x · z) traced through the exponential law Example
• The map x↦ x/(1+|x|) is a uniformly continuous homeomorphism from ℝ to (-1,1) whose inverse is not uniformly continuous Example
• Young's theorem integrates a Hölder function of unbounded variation against itself Example
• FALSE: a quotient of a Hausdorff space is Hausdorff False statement
• FALSE: all norms on a real vector space are equivalent False statement
• FALSE: the projections of a product are closed maps False statement
• A Cauchy sequence with a convergent subsequence converges, to that subsequence’s limit Lemma
• A compact subset of ℝ is closed and bounded Lemma
• A sequence has at most one limit Lemma
• Absolute real powers are Borel measurable and convex Lemma
• At a limit point of the domain a function has at most one limit Lemma
• Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page Lemma
• Every Cauchy sequence of reals is bounded Lemma
• Every convergent sequence is bounded Lemma
• Every convergent sequence is Cauchy Lemma
• Homogeneity and subadditivity of total variation Lemma
• If f has a finite limit at c then f is bounded on some punctured neighbourhood of c Lemma
• If for every ε > 0 some continuous g : X → ℝ satisfies | f(x) - g(x)| < ε for all x, then f is continuous; in particular a uniformly convergent series of continuous real functions has a continuous sum Lemma
• Low-frequency bound for the Weierstrass difference quotient Lemma
• Products converge uniformly when both factors converge uniformly and one limiting factor and one approximating family are uniformly bounded Lemma
• Refinement and tag-change estimates for Stieltjes sums Lemma
• ℝⁿ as the set of functions n → ℝ, and d₁, d₂, d_∞ are metrics on it Lemma
• Samuel function pseudometrics generate a uniformity coarser than the original one Lemma

…and 29 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

• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• T. Tao, Analysis I, 3rd ed. (standard reference, not scraped)
• Purdue University analysis notes: Ordered fields and absolute value (standard reference, not scraped)