The principle of mathematical induction

Depends on

• The natural numbers $\mathbb{N}$ (von Neumann)
• Inductive set
• The natural numbers exist: a smallest inductive set

Used by

• ∑_k<n+1C(n, k) = 2ⁿ, and ∑_k<n+1(-1)ᵏiotaC(n, k) = 0 for n ≥ 1 Corollary
• A finite row reduction from A to B is encoded by B=Eᵣ⋯ E₁A Corollary
• A Lebesgue measurable subgroup of (ℝⁿ,+) of positive measure is all of ℝⁿ Corollary
• A nonvanishing holomorphic function on such a domain has holomorphic roots of every positive order Corollary
• A polynomial ring in finitely many indeterminates over an integral domain is an integral domain Corollary
• A power-series sum is infinitely differentiable inside its radius and satisfies aₙ=f⁽ⁿ⁾(c)/ι(n!) at its centre Corollary
• A rational root of xᵏ = m is an integer: if k ≥ 1, m ∈ ℤ, x ∈ ℚ and xᵏ is the image of m, then x is the image of an integer Corollary
• Every nondegenerate interval of ℝ is uncountable Corollary
• Holomorphic functions of several variables are smooth and their complex derivatives are holomorphic Corollary
• If a prime p divides a finite product ∏_i<n aᵢ of integers then p ∣ aᵢ for some i < n; at n = 0 the product is 1 and the hypothesis cannot hold Corollary
• If V = ⨁_i<n Uᵢ with every Uᵢ finite-dimensional, then V is finite-dimensional and dim_F V = ∑_i<n dim_F Uᵢ; in particular dim_F(U ⊕ W) = dim_F U + dim_F W Corollary
• L(1/x)=-L(x), L(xⁿ)=nL(x), and in particular L(2ⁿ)=nL(2) Corollary
• Positive integers 4ᵃ m with m≡ 7 (mod 8) need four nonzero squares Corollary
• ℝ((t⁻¹)) has the nested interval property for lengths tending to 0 Corollary
• The a priori bound d(x^*, xₙ) ≤ qⁿ d(x₁,x₀)/(1-q) and the a posteriori bound d(x^*, xₙ₊₁) ≤ q d(xₙ₊₁,xₙ)/(1-q) Corollary
• The closed form for the volume of the unit n-ball Corollary
• The extended Euclidean algorithm: the same descent produces integers x, y with ax + by = gcd(a,b), so Bézout coefficients are computed and not merely shown to exist Corollary
• The higher-derivative form of the global Cauchy formula Corollary
• The modulus of a holomorphic function on a closed polydisc is bounded by its supremum on the distinguished boundary Corollary
• Γ(n+1)=n! for every natural number n Corollary
• ∏_j ≥ 0 (1 + (-1)ʲ/√j+2) has partial products tending to 0 although ∑_j ≥ 0 (-1)ʲ/√j+2 converges Counterexample
• A complete domain is necessary for sequential uniform boundedness Counterexample
• A nonnegative non-monotone sequence for which ∑ aₖ and ∑ 2ᵏ a_2ᵏ behave differently 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 disc algebra is unital and separating but not self-adjoint or dense Counterexample
• xₖ₊₁ = xₖ + 1/xₖ from x₁ = 1 has strictly decreasing consecutive gaps and diverges, so no uniform c < 1 exists Counterexample
• Balanced bracket words, defined by the recursive grammar Definition
• Cliques, independent sets, clique number and independence number Definition
• Diagonal lattice paths with steps U=(1,1) and D=(1,-1), and the height function Definition
• Epsilon-NFAs, finite epsilon closure, and extended word transition Definition
• Exponentiation of natural numbers, mⁿ, and its agreement with the integer power in ℝ Definition
• Finite sums and finite products of natural numbers, ∑_k<n aₖ and ∏_k<n aₖ in ℕ Definition
• Finite sums and finite products, by recursion Definition
• Grid partitions of a rectangle in ℝᵐ, their cells, refinements and mesh Definition
• Monotone lattice paths with steps E=(1,0) and N=(0,1) Definition
• Motzkin paths, Schröder paths, the Motzkin numbers Mₙ, the large Schröder numbers Rₙ, and their generating functions Definition
• Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences Definition
• Partition of [a,b] as a finite strictly increasing list a = t₀ < t₁ < … < tₙ = b, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions Definition
• Row space, column space, nullspace, row rank, column rank and matrix rank Definition
• Stay-put one-tape machines and their runs Definition

…and 296 more results.

Dependency tree · two levels

10 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

• Mathematical induction (Wikipedia) (standard reference, not scraped)
• Set-theoretic definition of natural numbers (Wikipedia) (standard reference, not scraped)
• B. Kaya, MATH 320 Set Theory (METU lecture notes) (standard reference, not scraped)
• Peano axioms (Wikipedia) (standard reference, not scraped)