Real analysis basics

Mathematics (B.Sc) · Calculus and analysis · free study notes for exams, placements and interviews

In short

Real analysis basics is the rigorous foundation of calculus: it fixes exactly what the real number system ℝ is (an ordered field that is *complete*), and uses that completeness to prove the facts calculus assumes — that bounded monotone sequences converge, that continuous functions on closed bounded intervals attain their bounds, that Riemann integrals exist. The subject is built on the supremum, the ε-δ/ε-N language, and the topology of ℝ (open, closed, compact, connected sets). It is a compulsory core paper in every B.Sc Mathematics CBCS/NEP syllabus and carries heavy weight in GATE MA, CSIR-UGC NET, IIT JAM MA, TIFR and ISI/CMI entrance. In jobs it underpins numerical analysis, optimisation and convergence proofs used by quant desks, actuarial and risk teams, and ML/data-science roles in India.

Explained

The real number system and completeness

ℝ as a complete ordered field: ℝ satisfies the field axioms (+, ×), the order axioms, and one extra axiom — completeness, which ℚ lacks. All of analysis follows from this one extra axiom.
Bounded above / below: S ⊆ ℝ is bounded above if ∃ M with x ≤ M ∀x ∈ S; M is an *upper bound*. Bounded = bounded above and below.
Supremum (least upper bound): sup S = the smallest upper bound. Characterisation used in proofs: s = sup S ⟺ (i) x ≤ s ∀x ∈ S, and (ii) ∀ε > 0 ∃x ∈ S with x > s − ε.
Infimum: inf S = greatest lower bound; inf S = −sup(−S). Both may or may not belong to S: for S = (0,1), inf = 0, sup = 1, neither attained; max/min exist only when attained.
Least Upper Bound (LUB) axiom: every non-empty subset of ℝ bounded above has a supremum in ℝ. Why ℚ fails: {x ∈ ℚ : x² < 2} is bounded above in ℚ but its least upper bound √2 ∉ ℚ.
Archimedean property: ∀x ∈ ℝ ∃n ∈ ℕ with n > x. Equivalently, ∀ε > 0 ∃n with 1/n < ε. Proved from the LUB axiom (if ℕ were bounded above, sup ℕ − 1 would be exceeded by some n).
Density of ℚ and of irrationals: between any two reals a < b there is a rational and an irrational. Hence ℚ is dense in ℝ; so is ℝ∖ℚ.
Equivalent forms of completeness (any one implies the rest): LUB property; every Cauchy sequence converges; monotone convergence theorem; Bolzano–Weierstrass; nested interval property; Heine–Borel; Dedekind cuts have no gaps.
Extended reals: by convention sup ∅ = −∞, sup of an unbounded-above set = +∞.
Absolute value and inequalities: |x| = max(x, −x); triangle inequality |x + y| ≤ |x| + |y|; |x − y| ≥ ||x| − |y||; AM ≥ GM ≥ HM; Cauchy–Schwarz (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²).

Countability and cardinality

Equinumerous: A ~ B if a bijection A → B exists. Finite, countably infinite (A ~ ℕ, i.e. denumerable), countable (finite or denumerable), uncountable.
Countable: ℕ, ℤ, ℚ (Cantor's diagonal enumeration of fractions), ℕ×ℕ, algebraic numbers, any subset of a countable set, a countable union of countable sets.
Uncountable: ℝ, any interval (a,b) with a < b, the irrationals, the Cantor set, the power set 𝒫(ℕ).
Cantor's diagonal argument: assume (0,1) is listed as decimal expansions x₁, x₂, …; build y differing from xₙ in the nth digit — y ∈ (0,1) is not in the list. Hence ℝ is uncountable.
Cardinal facts: |ℕ| = ℵ₀, |ℝ| = c = 2^ℵ₀ > ℵ₀. Continuum hypothesis (c = ℵ₁) is independent of ZFC — neither provable nor refutable, a standard viva point.
Consequence: "almost all" reals are irrational and transcendental; the transcendentals are uncountable because algebraic numbers are countable.

Topology of ℝ

Neighbourhood: an open interval (x − δ, x + δ) around x. Open set: every point has a neighbourhood inside the set. Closed set: complement is open ⟺ contains all its limit points.
Limit (cluster/accumulation) point: p is a limit point of S if every neighbourhood of p contains a point of S other than p. Isolated point: in S but not a limit point. Derived set S′ = set of limit points; closure S̄ = S ∪ S′; interior S°; boundary ∂S = S̄ ∖ S°.
Unions of arbitrarily many open sets are open; *finite* intersections of open sets are open — infinite ones need not be: ∩ₙ(−1/n, 1/n) = {0}, closed. Dually for closed sets.
∅ and ℝ are both open and closed; structure theorem: every open set in ℝ is a countable union of disjoint open intervals.
Compact set: every open cover has a finite subcover. Heine–Borel theorem: K ⊆ ℝ is compact ⟺ K is closed and bounded. (True in ℝⁿ; fails in general metric spaces, where compact ⟺ complete + totally bounded.)
Bolzano–Weierstrass: every bounded infinite subset of ℝ has a limit point; equivalently every bounded sequence has a convergent subsequence.
Nested interval property: if Iₙ = [aₙ, bₙ] with I₁ ⊇ I₂ ⊇ …, then ∩Iₙ ≠ ∅; if additionally bₙ − aₙ → 0 the intersection is a single point. Closedness is essential: ∩(0, 1/n) = ∅.
Dense, nowhere dense, perfect: S is dense in ℝ if S̄ = ℝ; perfect if S = S′ (closed with no isolated points) — e.g. [0,1] and the Cantor set. The Cantor set is uncountable, compact, perfect, nowhere dense, of Lebesgue measure zero.
Connected subsets of ℝ are exactly the intervals — this is what makes the Intermediate Value Theorem true.

Sequences, Cauchy criterion and limsup

Convergence (ε–N): xₙ → L if ∀ε > 0 ∃N ∈ ℕ such that n ≥ N ⟹ |xₙ − L| < ε. A convergent sequence is bounded and has a unique limit.
Monotone convergence theorem: a monotone increasing sequence bounded above converges to sup{xₙ}; this *is* completeness in sequence form.
Cauchy sequence: ∀ε > 0 ∃N with |xₙ − xₘ| < ε ∀m, n ≥ N. Cauchy completeness of ℝ: a real sequence converges ⟺ it is Cauchy. Advantage: convergence can be tested without knowing the limit. ℚ is not complete — the decimal truncations of √2 are Cauchy in ℚ with no rational limit.
Subsequential limits: limsup xₙ = inf_n sup_{k≥n} x_k, liminf xₙ = sup_n inf_{k≥n} x_k. These always exist in [−∞, ∞]; xₙ → L ⟺ limsup = liminf = L. limsup is the largest subsequential limit.
Sandwich, algebra of limits, Cesàro mean, ratio test for sequences — standard tools; detailed convergence tests for series belong to the sequences-and-series topic.

Continuity, uniform continuity and the big theorems

Continuity at a point (ε–δ): f is continuous at c if ∀ε > 0 ∃δ > 0 such that |x − c| < δ ⟹ |f(x) − f(c)| < ε. Sequential form: xₙ → c ⟹ f(xₙ) → f(c).
Global characterisation: f : ℝ → ℝ is continuous ⟺ the preimage of every open set is open. Continuous image of a compact set is compact; of a connected set is connected.
Boundedness and extreme value theorems: f continuous on a closed bounded [a, b] is bounded and attains its supremum and infimum. Both fail on open intervals: f(x) = 1/x on (0,1).
Intermediate Value Theorem: f continuous on [a, b], f(a)·f(b) < 0 ⟹ ∃c ∈ (a, b) with f(c) = 0; basis of the bisection method.
Uniform continuity: ∀ε > 0 ∃δ > 0 working for *all* pairs: |x − y| < δ ⟹ |f(x) − f(y)| < ε. δ depends only on ε, not on the point. Uniform ⟹ continuous, not conversely.
Heine's theorem: continuous on a compact set ⟹ uniformly continuous. Hence x² is uniformly continuous on [0, 5] but not on ℝ; sin(1/x) is continuous but not uniformly continuous on (0, 1).
Lipschitz condition: |f(x) − f(y)| ≤ K|x − y|. Lipschitz ⟹ uniformly continuous ⟹ continuous, and the implications are strict (√x on [0,1] is uniformly continuous, not Lipschitz).
Types of discontinuity: removable (limit exists ≠ f(c)); jump/first kind (both one-sided limits exist, unequal); second kind (a one-sided limit fails to exist, e.g. sin(1/x) at 0). A monotone function has at most countably many discontinuities, all of the first kind.
Pathologies worth quoting: Dirichlet function 1_ℚ is nowhere continuous; Thomae's function (1/q at p/q in lowest terms, 0 at irrationals) is continuous exactly at the irrationals; |x| is continuous but not differentiable at 0.

Riemann integration — the analytic criterion

Partition and sums: for P = {a = x₀ < … < xₙ = b}, L(P, f) = Σ mᵢΔxᵢ, U(P, f) = Σ MᵢΔxᵢ with mᵢ, Mᵢ the inf and sup on [xᵢ₋₁, xᵢ].
Refining a partition increases L and decreases U; lower integral ∫_ = sup_P L ≤ upper integral ∫¯ = inf_P U.
Riemann integrable: f bounded on [a, b] is integrable if ∫_ = ∫¯. Riemann's criterion: integrable ⟺ ∀ε > 0 ∃P with U(P, f) − L(P, f) < ε.
Sufficient classes: continuous on [a,b]; monotone on [a,b]; bounded with finitely many (indeed, a measure-zero set of) discontinuities. Lebesgue's criterion: a bounded f is Riemann integrable ⟺ its set of discontinuities has measure zero.
Not integrable: 1_ℚ on [0,1] (L = 0, U = 1); unbounded functions (handled as improper integrals).
Fundamental Theorem of Calculus: (i) if f is integrable and F(x) = ∫ₐˣ f, then F is continuous, and F′(c) = f(c) wherever f is continuous; (ii) if f = g′ with g′ Riemann integrable, ∫ₐᵇ f = g(b) − g(a). Mean value theorem for integrals: f continuous ⟹ ∃c with ∫ₐᵇ f = f(c)(b − a).
Limitation of the Riemann integral: it handles limits badly (pointwise limits of integrable functions need not be integrable) — the reason the Lebesgue integral is introduced in M.Sc courses.

Metric spaces (the generalisation examined in GATE/NET)

Metric: d : X × X → [0, ∞) with d(x,y) = 0 ⟺ x = y, symmetry, and triangle inequality d(x,z) ≤ d(x,y) + d(y,z). ℝ with d(x,y) = |x − y| is the model case.
Examples: discrete metric (d = 0 or 1); ℝⁿ with Euclidean, taxicab (ℓ¹) and sup (ℓ^∞) metrics — these three are equivalent (same open sets, same convergent sequences).
Open balls B(x, r) generate the topology; definitions of limit point, closure, completeness, compactness and continuity carry over verbatim.
Complete metric space: every Cauchy sequence converges — ℝ, ℝⁿ, C[a,b] with the sup metric are complete; ℚ and (0,1) are not.
Banach fixed point theorem: a contraction (d(Tx,Ty) ≤ k·d(x,y), 0 ≤ k < 1) on a non-empty complete metric space has a unique fixed point, found by iteration — the convergence proof behind Newton–Raphson, Picard's existence theorem for ODEs and many numerical schemes.

Key points

Completeness axiom — every non-empty set bounded above has a supremum in ℝ; the single property separating ℝ from ℚ.
Supremum test — s = sup S ⟺ s is an upper bound and every s − ε is exceeded by some element of S.
Archimedean property — ℕ is unbounded in ℝ; equivalently 1/n → 0.
Density — every open interval contains both rationals and irrationals.
Countable vs uncountable — ℚ is countable, ℝ and the irrationals are not (Cantor's diagonal argument).
Open set — every point is interior; closed set — contains all its limit points; both ∅ and ℝ are each.
Heine–Borel — in ℝ (and ℝⁿ), compact ⟺ closed and bounded.
Bolzano–Weierstrass — every bounded sequence has a convergent subsequence; every bounded infinite set has a limit point.
Nested interval property — a decreasing chain of closed bounded intervals has non-empty intersection.
Cauchy criterion — in ℝ, convergent ⟺ Cauchy; lets you prove convergence without the limit.
limsup / liminf — always exist in the extended reals; equal ⟺ the sequence converges.
Monotone convergence — bounded monotone sequences converge, to the sup (or inf).
Uniform continuity — one δ for all points; on a compact set continuity is automatically uniform (Heine).
Lipschitz ⟹ uniformly continuous ⟹ continuous — strict chain of implications.
Extreme value theorem — continuity on a closed bounded interval gives boundedness and attained max/min.
IVT — rests on intervals being the connected subsets of ℝ; gives the bisection root-finding method.
Riemann criterion — integrable ⟺ U(P,f) − L(P,f) can be made arbitrarily small; 1_ℚ fails it.
Banach contraction principle — a contraction on a complete space has exactly one fixed point, reached by iteration.

Formulas and examples

sup S = least upper bound of S; inf S = greatest lower bound. ε-characterisation: ∀ε > 0, ∃x ∈ S with x > sup S − ε.
Convergence: xₙ → L ⟺ ∀ε > 0 ∃N ∈ ℕ : n ≥ N ⟹ |xₙ − L| < ε. (ε is a positive real tolerance; N a natural number index.)
Cauchy: ∀ε > 0 ∃N : m, n ≥ N ⟹ |xₙ − xₘ| < ε.
limsup xₙ = lim_{n→∞} ( sup_{k ≥ n} x_k ); liminf xₙ = lim_{n→∞} ( inf_{k ≥ n} x_k ).
Continuity at c: ∀ε > 0 ∃δ > 0 : |x − c| < δ ⟹ |f(x) − f(c)| < ε. (δ may depend on both ε and c.)
Uniform continuity on A: ∀ε > 0 ∃δ > 0 : x, y ∈ A, |x − y| < δ ⟹ |f(x) − f(y)| < ε. (δ depends on ε only.)
Lipschitz: |f(x) − f(y)| ≤ K|x − y|, K ≥ 0 the Lipschitz constant. If f′ exists and |f′| ≤ K on an interval, f is Lipschitz with that K.
Darboux sums: L(P,f) = Σᵢ₌₁ⁿ mᵢ(xᵢ − xᵢ₋₁), U(P,f) = Σᵢ₌₁ⁿ Mᵢ(xᵢ − xᵢ₋₁); integrable ⟺ inf U = sup L.
Triangle inequality: |x + y| ≤ |x| + |y|; metric form d(x,z) ≤ d(x,y) + d(y,z).
Contraction mapping: d(Tx, Ty) ≤ k·d(x, y), 0 ≤ k < 1 ⟹ unique fixed point x* with d(xₙ, x*) ≤ kⁿ d(x₀, x*) and error bound d(xₙ, x*) ≤ kⁿ/(1 − k) · d(x₁, x₀).
Σᵢ₌₁ⁿ i = n(n+1)/2; Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6 — needed for Riemann sums from first principles.

Example 1 — Supremum and infimum of a set. Let S = {1 − 1/n : n ∈ ℕ} = {0, 1/2, 2/3, 3/4, …}.

Every element is < 1, so 1 is an upper bound.
Is any smaller number an upper bound? Given ε > 0, choose n > 1/ε by the Archimedean property; then 1 − 1/n > 1 − ε. So no number below 1 bounds S.
sup S = 1, not attained, so max S does not exist. The smallest element is 1 − 1/1 = 0, so inf S = min S = 0.

Example 2 — ε–N proof of a limit. Show xₙ = (3n + 2)/(n + 4) → 3.

|xₙ − 3| = |(3n + 2 − 3n − 12)/(n + 4)| = 10/(n + 4).
Require 10/(n + 4) < ε ⟹ n + 4 > 10/ε ⟹ n > 10/ε − 4.
Take N = ⌈10/ε⌉ (which exceeds 10/ε − 4). For ε = 0.01: N = 1000, and at n = 1000, |x₁₀₀₀ − 3| = 10/1004 ≈ 0.00996 < 0.01. ✓ Limit is 3.

Example 3 — Continuous but not uniformly continuous. f(x) = 1/x on (0, 1].

Take xₙ = 1/n, yₙ = 1/(2n). Then |xₙ − yₙ| = 1/(2n) → 0, but |f(xₙ) − f(yₙ)| = |n − 2n| = n → ∞.
So for ε = 1 no δ works: f is not uniformly continuous on (0,1], though it is continuous at every point. On [1/2, 1] it is uniformly continuous — there |f′(x)| = 1/x² ≤ 4, so it is Lipschitz with K = 4 (consistent with Heine's theorem on the compact set).

Example 4 — Riemann integrability from first principles. f(x) = x on [0, 2], with the uniform partition xᵢ = 2i/n, Δx = 2/n.

f increasing ⟹ mᵢ = x_{i−1} = 2(i−1)/n, Mᵢ = xᵢ = 2i/n.
U(P,f) = Σᵢ₌₁ⁿ (2i/n)(2/n) = (4/n²)·n(n+1)/2 = 2(n+1)/n = 2 + 2/n.
L(P,f) = Σᵢ₌₁ⁿ (2(i−1)/n)(2/n) = (4/n²)·(n−1)n/2 = 2 − 2/n.
U − L = 4/n → 0, so Riemann's criterion holds. For n = 100: U = 2.02, L = 1.98, U − L = 0.04. Both sums → 2 = ∫₀² x dx = 2²/2. ✓

Example 5 — Non-integrable function and a contraction.

(a) f = 1_ℚ on [0,1]: every subinterval contains a rational and an irrational, so mᵢ = 0, Mᵢ = 1 for every partition ⟹ L = 0, U = 1, U − L = 1 ↛ 0. Not Riemann integrable.
(b) T(x) = (x + 2/x)/2 on [1, 2] (Newton's iteration for √2). T′(x) = (1 − 2/x²)/2, and on [1,2] |T′| ≤ 1/2, so T is a contraction with k = 1/2 on the complete space [1,2]. Starting x₀ = 1: x₁ = 1.5, x₂ = 1.4166667, x₃ = 1.4142157, x₄ ≈ 1.4142136 — converging to the unique fixed point √2 ≈ 1.4142136.

Property	On (0, 1) — open	On [0, 1] — closed, bounded
Compact (Heine–Borel)	No	Yes
Complete as a metric space	No (xₙ = 1/n is Cauchy, no limit inside)	Yes
Continuous f bounded?	Not necessarily (1/x)	Always
Max/min attained?	Not necessarily	Always
Continuity ⟹ uniform continuity?	No (sin(1/x))	Yes (Heine)

Concept	Pointwise statement	Uniform / global statement
Continuity	δ depends on ε and c	δ depends on ε only
Convergence of sequences	per-point limit	uniform convergence on a set
Typical counterexample	1/x on (0,1)	—

Where it is used

Quantitative finance and risk, India: convergence and stability proofs for Monte Carlo and binomial pricing engines at quant desks and risk teams (ICICI Securities, HDFC Bank treasury, Goldman Sachs Bengaluru, Morgan Stanley Mumbai); analysis questions are standard in quant interviews.
Actuarial work: IAI (Institute of Actuaries of India) CM/CS subjects assume limits, convergence of interest-accumulation series and integrability of density functions.
Numerical computing and engineering software: error bounds, convergence of bisection (IVT), Newton–Raphson and fixed-point iteration (contraction principle) in solvers at ISRO, BARC, DRDO, Ansys/Siemens India R&D.
Machine learning and optimisation roles: convexity, compactness and continuity guarantee that minima exist; gradient-descent convergence proofs use Cauchy sequences and Lipschitz constants.
Academia, teaching and competitive exams: core paper for B.Sc/M.Sc, and a major section of GATE MA, CSIR-UGC NET Mathematical Sciences, IIT JAM MA, ISI/CMI entrance and UPSC Mathematics optional.
Signal processing and statistics: measure-zero and almost-everywhere arguments justify sampling theory and the convergence of estimators used in data-analytics roles.

Interview questions

Q: What exactly makes ℝ different from ℚ? A: ℝ is complete — every non-empty subset bounded above has a least upper bound in ℝ. ℚ fails this: {x ∈ ℚ : x² < 2} is bounded above but has no rational least upper bound.
Q: Give a set whose supremum is not a member of it. A: (0, 1) has sup = 1 ∉ the set; equivalently {1 − 1/n : n ∈ ℕ} has sup = 1, never attained.
Q: Is ℚ countable? Are the irrationals? A: ℚ is countable (it can be enumerated as a countable union of countable sets). The irrationals are uncountable — if they were countable, ℝ = ℚ ∪ (ℝ∖ℚ) would be countable.
Q: Difference between continuity and uniform continuity? A: For continuity δ may depend on the point; for uniform continuity a single δ works everywhere on the set. 1/x on (0,1) is continuous but not uniformly continuous.
Q: When does continuity automatically give uniform continuity? A: On a compact set — Heine's theorem. In ℝ that means a closed bounded set, by Heine–Borel.
Q: State Heine–Borel and give a set that is closed but not compact. A: In ℝⁿ, compact ⟺ closed and bounded. [0, ∞) is closed but unbounded, hence not compact (the cover {(−1, n)} has no finite subcover).
Q: Why is the Cauchy criterion useful? A: It certifies convergence using only the terms of the sequence, without knowing the limit — essential when the limit is the object being constructed, as in completions and numerical iterations.
Q: Is every Cauchy sequence convergent? A: In a complete space such as ℝ, yes. Not in general: the decimal truncations of √2 are Cauchy in ℚ but have no limit in ℚ.
Q: Give a bounded function that is not Riemann integrable. A: The Dirichlet function 1_ℚ on [0,1]: every lower sum is 0 and every upper sum is 1, so the lower and upper integrals differ.
Q: Can a function be discontinuous at every rational and continuous at every irrational? A: Yes — Thomae's function, f(p/q) = 1/q in lowest terms and 0 at irrationals, is continuous exactly at the irrationals.
Q: What does Bolzano–Weierstrass say and where is it used? A: Every bounded sequence in ℝ has a convergent subsequence; it is used to prove the extreme value theorem and the existence of limit points in compactness arguments.
Q: Why does the bisection method always work for a sign change? A: By the Intermediate Value Theorem, which holds because intervals are exactly the connected subsets of ℝ; halving gives nested intervals whose lengths → 0, so they intersect in the root.
Q: Does an infinite intersection of open sets stay open? A: No — ∩ₙ₌₁^∞ (−1/n, 1/n) = {0}, which is closed. Only finite intersections preserve openness.

Common mistakes

Confusing supremum with maximum — writing "max of (0,1) is 1" instead of sup = 1 with no maximum.
Proving sup S only by showing it is an upper bound, and skipping the *least*-ness (the ε-step).
Writing the ε–δ or ε–N quantifiers in the wrong order, or letting N depend on n, or choosing δ that secretly depends on x in a uniform-continuity proof.
Assuming continuous ⟹ uniformly continuous, or bounded ⟹ convergent; and assuming differentiable ⟹ Lipschitz on an open interval (tan x on (−π/2, π/2) is not).
Using Heine–Borel in a general metric space: closed and bounded does not imply compact there (the unit ball of an infinite-dimensional space is the standard counterexample).
Treating "closed" as "not open" — ∅ and ℝ are both; [0,1) is neither.
Claiming a bounded function with infinitely many discontinuities cannot be integrable; what matters is whether the discontinuity set has measure zero (Lebesgue's criterion).
Computing a Riemann integral from Darboux sums but dropping the (n−1)n/2 versus n(n+1)/2 distinction between lower and upper sums.