Editing Real Analysis (section)

== <span style="color: #FFFFFF;">Understanding</span> ==
Real analysis develops four fundamental concepts with increasing rigor:

'''Limits and the completeness of ℝ''': The rationals ℚ have "holes" — the sequence 1, 1.4, 1.41, 1.414, ... converges to √2, which isn't rational. The reals are constructed to fill these holes: Dedekind cuts or Cauchy completion. The least upper bound property (every bounded-above non-empty set has a supremum) is the key axiom. Consequences: the Bolzano-Weierstrass theorem (every bounded sequence in ℝ has a convergent subsequence), the Heine-Borel theorem (a subset of ℝ^n is compact iff it is closed and bounded).

'''Continuity and its consequences''': The epsilon-delta definition captures the intuition that a function is continuous if nearby inputs give nearby outputs, without any reference to "infinitesimals." Key theorems: the Extreme Value Theorem (a continuous function on a compact set attains its max and min — used in optimization), the Intermediate Value Theorem (a continuous function on [a,b] that takes values c₁ and c₂ takes all values between them — used to prove the existence of roots), and Uniform Continuity on compact sets.

'''Differentiation and integration''': The Fundamental Theorem of Calculus (FTC) is the deepest result of basic analysis: differentiation and integration are inverse operations. But the theorem requires precise formulation: f must be integrable, F = ∫ₐˣ f(t)dt is differentiable with F' = f (FTC Part 1); if F' = f on [a,b] then ∫ₐᵇ f = F(b) - F(a) (FTC Part 2). The mean value theorem (f(b)-f(a) = f'(c)(b-a) for some c in (a,b)) underlies error bounds for approximations.

'''Series and the problem of interchanging limits''': When can we integrate or differentiate a series term-by-term? Uniform convergence is the key: if fₙ → f uniformly and each fₙ is continuous (resp. integrable), then f is continuous (resp. integrable) and ∫fₙ → ∫f. Point-wise convergence is not enough: the classic example of functions fₙ(x) = xⁿ on [0,1] converges pointwise to a discontinuous function. Uniform convergence distinguishes safe limit-interchange from dangerous ones.
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