Editing Real Analysis (section)

== <span style="color: #FFFFFF;">Remembering</span> ==
* '''Real analysis''' — The rigorous mathematical study of the real numbers, sequences, series, limits, continuity, differentiation, and integration.
* '''Real numbers (ℝ)''' — The complete ordered field; characterized uniquely (up to isomorphism) by the least upper bound property.
* '''Least upper bound (supremum)''' — The smallest upper bound of a set; the completeness axiom: every non-empty set with an upper bound has a supremum in ℝ.
* '''Completeness of ℝ''' — Every Cauchy sequence in ℝ converges (to a real number); no "holes" in ℝ (unlike ℚ).
* '''Sequence''' — A function ℕ → ℝ; the foundation of limit theory.
* '''Limit of a sequence''' — L is the limit of (aₙ) if: for every ε > 0, ∃ N such that n > N ⟹ |aₙ - L| < ε.
* '''Cauchy sequence''' — A sequence where |aₙ - aₘ| → 0 as n,m → ∞; converges iff ℝ is complete.
* '''Epsilon-delta definition (continuity)''' — f is continuous at c if: ∀ε>0, ∃δ>0: |x-c|<δ ⟹ |f(x)-f(c)|<ε.
* '''Uniform continuity''' — δ depends only on ε, not on the point c; stronger than pointwise continuity.
* '''Differentiability''' — f'(x) = lim_{h→0} [f(x+h)-f(x)]/h; requires the limit to exist.
* '''Riemann integral''' — Defined via upper and lower sums; the standard calculus integral.
* '''Lebesgue integral''' — A more powerful integral based on measure theory; integrates functions Riemann cannot.
* '''Series''' — Sum of a sequence; Σaₙ; convergence is the limit of partial sums.
* '''Uniform convergence''' — A sequence of functions fₙ → f uniformly if: ∀ε>0, ∃N: ∀x, n>N ⟹ |fₙ(x)-f(x)|<ε.
* '''Power series''' — A series Σ aₙxⁿ; converges in a disk; the basis of Taylor series.
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