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Real Analysis — Foundations

Rigorous treatment of limits, continuity, and the real number system

E
epsilon_delta 24 terms Feb 6, 2026
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Terms 24

1
Completeness Axiom
Every nonempty set of real numbers bounded above has a least upper bound (supremum)
2
Supremum (least upper bound)
Smallest upper bound of a set; sup S = smallest M such that x ≤ M for all x in S
3
Infimum (greatest lower bound)
Largest lower bound of a set; inf S = largest m such that m ≤ x for all x in S
4
ε–δ Definition of Limit
lim_{x→a} f(x) = L if ∀ε>0, ∃δ>0 such that 0<|x−a|<δ ⟹ |f(x)−L|<ε
5
Continuity (ε–δ)
f continuous at a if ∀ε>0, ∃δ>0 such that |x−a|<δ ⟹ |f(x)−f(a)|<ε
6
Uniform Continuity
Same δ works for all x in domain; stronger than pointwise continuity; compact domains guarantee it
7
Cauchy Sequence
Sequence where terms get arbitrarily close: ∀ε>0, ∃N such that m,n>N ⟹ |a_m−a_n|<ε
8
Completeness (sequences)
Every Cauchy sequence of real numbers converges; equivalent to completeness axiom
9
Bolzano-Weierstrass Theorem
Every bounded sequence has a convergent subsequence
10
Heine-Borel Theorem
Subset of ℝⁿ is compact iff it is closed and bounded
11
Intermediate Value Theorem
If f continuous on [a,b] and f(a) < k < f(b), then ∃c ∈ (a,b) with f(c) = k
12
Extreme Value Theorem
Continuous function on compact set attains its maximum and minimum
13
Mean Value Theorem
If f differentiable on (a,b), ∃c with f'(c) = (f(b)−f(a))/(b−a)
14
Rolle's Theorem
If f(a) = f(b) and f differentiable on (a,b), then ∃c with f'(c) = 0
15
Taylor's Theorem with Remainder
f(x) = Σ f^(n)(a)/n! (x−a)^n + R_n; remainder R_n bounds approximation error
16
Uniform Convergence
Sequence of functions f_n → f uniformly if sup|f_n(x)−f(x)| → 0; preserves continuity, integration
17
Pointwise vs Uniform Convergence
Pointwise: each x converges separately; uniform: rate of convergence same across domain
18
Riemann Integral
Limit of Riemann sums; ∫[a,b] f = sup{L(P,f)} = inf{U(P,f)} for integrable f
19
Lebesgue Integral
More general integral; integrates over measure of function's level sets; handles more functions
20
Measure Zero
Set coverable by intervals of arbitrarily small total length; Riemann-integrable iff discontinuities have measure zero
21
Open Set
Every point has neighborhood contained in set; arbitrary unions and finite intersections of open sets are open
22
Closed Set
Contains all its limit points; complement of open set; closed under arbitrary intersections
23
Dense Set
Every open interval contains a point of the set; rationals are dense in reals
24
Archimedean Property
For any x ∈ ℝ, there exists n ∈ ℕ with n > x; ℝ has no infinitely large or small elements