ProofZoom entries are concise expository articles organized by area and subarea. Each entry is available in web form for online reading and, where provided, as a PDF for download and offline use.
We construct $\mathbb{N}$ inside ZF set theory as the smallest inductive set, define it as the intersection of all inductive sets, and establish its fundamental properties. We conclude with remarks on categoricity and nonstandard models.
This entry develops the geometric meaning of tangent direction for complex-valued curves, explains how angles between curves may be expressed through arguments of complex numbers, and shows how this leads naturally to conformality. Particular attention is given to the condition $z'(t_0) ≠ 0$, whose meaning in the complex setting is subtler than in the real case.
We define the complex line integral of a continuous function on a rectifiable curve via Riemann-type sums and prove that the definition is well-posed, independent of the choice of tags. The existence of the integral rests on a fundamental balance between analytic control, via uniform continuity, and geometric control, via finite length. We then show that for piecewise C¹ curves this definition agrees with the classical parametrized integral from a to b of f(γ(t))γ'(t) dt, and derive the fundamental estimate that the magnitude of the integral over C is bounded by the maximum of |f(z)| on C times the length of C. Finally, we establish the length formula for piecewise C¹ curves and interpret the line integral as a construction combining analytic, geometric, and algebraic features.
We present a proof of Zorn’s Lemma based on Lewin’s notion of a conforming set. The proof constructs chains recursively using a choice function and shows that all such constructions are forced to agree on their initial segments. Their union therefore forms a single maximal construction, which the choice function paradoxically compels to extend further, yielding a contradiction.
Cauchy’s theorem asserts that the integral of a holomorphic function around a closed curve vanishes. Beyond this striking phenomenon lies a deeper truth: the theorem reflects a fundamental property of the domain itself. In simply connected regions, holomorphic functions behave as global derivatives, and their integrals detect no circulation. Conversely, any failure of such behavior reveals the presence of a hole. Thus complex integration encodes the topology of the plane.
Cauchy’s integral formula expresses the value of an analytic function at a point in terms of its values on a surrounding curve. The proof proceeds by isolating the contribution of a single interior point: an auxiliary function is constructed so that its contour integral vanishes, either by boundedness near the point or by a sharper decay condition. This reveals that the entire integral is governed by the local behavior of the integrand near that point. Consequently, analytic functions are completely determined by their boundary values, possess derivatives of all orders, and admit power series expansions.
We prove that Zorn’s Lemma implies the Well-Ordering Theorem, and show how a well-ordering yields the Axiom of Choice. The proof constructs a well-ordering by extending partial well-orderings to a maximal one, which must include every element of the set. Once such a global order is available, each set in a family acquires a canonical choice as its least element. Together with the converse implication, this establishes the equivalence of the Axiom of Choice, Zorn’s Lemma, and the Well-Ordering Theorem, revealing a common principle underlying selection, maximality, and order.
Singularities encode precise structural information about analytic functions. We classify isolated singularities using boundedness via Riemann's removable singularity theorem, explain how Taylor and Laurent expansions arise from this viewpoint, and show why residues are the part of a singularity detected by contour integration.