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Real Analysis

Rigorous study of sequences, series, continuity, differentiation, integration (Riemann and Lebesgue), measure theory, and function spaces at the CSIR NET level.

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01 Sequences & Series

Convergence of real sequences, Cauchy criterion, series convergence tests, and absolute vs conditional convergence.

◆ Convergence of Sequences
Definition A sequence $(a_n)$ in $\mathbb{R}$ converges to $L \in \mathbb{R}$ if for every $\varepsilon > 0$, there exists $N \in \mathbb{N}$ such that $|a_n - L| < \varepsilon$ for all $n \geq N$.
Theorem (Bolzano-Weierstrass) Every bounded sequence in $\mathbb{R}$ has a convergent subsequence.
Cauchy Criterion A sequence $(a_n)$ in $\mathbb{R}$ converges if and only if it is a Cauchy sequence: for every $\varepsilon > 0$, there exists $N$ such that $|a_m - a_n| < \varepsilon$ for all $m, n \geq N$. This is equivalent to the completeness of $\mathbb{R}$.
📝 Example
Show that $a_n = \left(1 + \frac{1}{n}\right)^n$ converges.
The sequence is monotonically increasing and bounded above by $3$. By the Monotone Convergence Theorem, it converges. Its limit is $e = 2.71828\ldots$
◆ Series Convergence Tests
Comparison Test If $0 \leq a_n \leq b_n$ for all $n$ and $\sum b_n$ converges, then $\sum a_n$ converges.
Root & Ratio Tests For $\sum a_n$ with $a_n > 0$: if $\limsup_{n\to\infty} \sqrt[n]{a_n} = L$, then the series converges if $L < 1$ and diverges if $L > 1$. Similarly for the ratio test with $\lim \frac{a_{n+1}}{a_n}$.
Abel's & Dirichlet's Tests Dirichlet: If $(a_n)$ is monotonically decreasing to $0$ and the partial sums of $\sum b_n$ are bounded, then $\sum a_n b_n$ converges.
Abel: If $\sum b_n$ converges and $(a_n)$ is monotone and bounded, then $\sum a_n b_n$ converges.
📝 Example
Test the convergence of $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^p}$ for $p > 0$.
By the Alternating Series Test (Leibniz), since $\frac{1}{n^p} \to 0$ monotonically for $p > 0$, the series converges. It converges absolutely if and only if $p > 1$ (by the $p$-series test).
02 Continuity & Differentiability

Uniform continuity, mean value theorems, Taylor's theorem, and properties of differentiable functions on $\mathbb{R}$.

◆ Uniform Continuity
Definition A function $f: A \to \mathbb{R}$ is uniformly continuous on $A$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(x) - f(y)| < \varepsilon$ whenever $|x - y| < \delta$ for all $x, y \in A$.
Theorem (Heine-Cantor) Every continuous function on a compact set is uniformly continuous.

Key distinction: for uniform continuity, $\delta$ depends only on $\varepsilon$, not on the point. The function $f(x) = \sin(x^2)$ is continuous on $\mathbb{R}$ but not uniformly continuous, since oscillations become arbitrarily rapid.

📝 Example
Prove that $f(x) = \frac{1}{x}$ is not uniformly continuous on $(0,1)$.
Take $x_n = \frac{1}{n}$ and $y_n = \frac{1}{n+1}$. Then $|x_n - y_n| = \frac{1}{n(n+1)} \to 0$, but $|f(x_n) - f(y_n)| = 1$ for all $n$. Hence $f$ is not uniformly continuous on $(0,1)$.
◆ Mean Value Theorems
Lagrange's Mean Value Theorem If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, there exists $c \in (a,b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.
Cauchy's Mean Value Theorem If $f, g$ are continuous on $[a,b]$ and differentiable on $(a,b)$ with $g'(x) \neq 0$, there exists $c \in (a,b)$ such that $\frac{f'(c)}{g'(c)} = \frac{f(b) - f(a)}{g(b) - g(a)}$.
Taylor's Theorem If $f$ is $n$-times differentiable at $a$, then $f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k + R_n(x)$, where the Lagrange remainder is $R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ for some $c$ between $a$ and $x$.
03 Riemann & Lebesgue Integration

Riemann integrability criteria, improper integrals, and the Lebesgue integral with its powerful convergence theorems.

◆ Riemann Integration
Definition A bounded function $f:[a,b] \to \mathbb{R}$ is Riemann integrable if $\sup_P L(f,P) = \inf_P U(f,P)$, where $L$ and $U$ are the lower and upper Darboux sums over partitions $P$.
Lebesgue's Criterion A bounded function $f:[a,b] \to \mathbb{R}$ is Riemann integrable if and only if the set of discontinuities of $f$ has Lebesgue measure zero.

Key facts: Every continuous function is Riemann integrable. Every monotone function on $[a,b]$ is Riemann integrable (it has at most countably many discontinuities).

◆ Lebesgue Integration
Definition For a measurable function $f: X \to [0, \infty]$, the Lebesgue integral is defined as $\int f \, d\mu = \sup\left\{ \int \phi \, d\mu : 0 \leq \phi \leq f, \, \phi \text{ simple}\right\}$.
Monotone Convergence Theorem (MCT) If $0 \leq f_1 \leq f_2 \leq \cdots$ are measurable and $f_n \uparrow f$ pointwise, then $\int f_n \, d\mu \uparrow \int f \, d\mu$.
Dominated Convergence Theorem (DCT) If $f_n \to f$ pointwise a.e. and $|f_n| \leq g$ a.e. for some integrable $g$, then $\int f_n \, d\mu \to \int f \, d\mu$.
Fatou's Lemma If $f_n \geq 0$ are measurable, then $\int \liminf f_n \, d\mu \leq \liminf \int f_n \, d\mu$.
📝 Example
Compute $\lim_{n\to\infty} \int_0^1 \frac{n x}{1 + n^2 x^2} \, dx$ using DCT.
For each fixed $x > 0$, $\frac{nx}{1+n^2x^2} \to 0$ as $n \to \infty$. Also $\left|\frac{nx}{1+n^2x^2}\right| \leq \frac{1}{2}$ (by AM-GM). By DCT, the limit is $\int_0^1 0 \, dx = 0$.
04 Measure Theory

Sigma-algebras, measurable sets, measurable functions, and construction of Lebesgue measure on $\mathbb{R}$.

◆ Measurable Sets & Lebesgue Measure
Definition A $\sigma$-algebra $\mathcal{F}$ on a set $X$ is a collection of subsets containing $\emptyset$, closed under complements and countable unions. The pair $(X, \mathcal{F})$ is a measurable space.
Lebesgue Outer Measure For $A \subseteq \mathbb{R}$, $m^*(A) = \inf\left\{\sum_{k=1}^{\infty} \ell(I_k) : A \subseteq \bigcup_{k=1}^{\infty} I_k\right\}$, where the infimum is over countable covers by open intervals $I_k$ with lengths $\ell(I_k)$.
Caratheodory's Criterion A set $E \subseteq \mathbb{R}$ is Lebesgue measurable if for every $A \subseteq \mathbb{R}$: $m^*(A) = m^*(A \cap E) + m^*(A \cap E^c)$.

Properties of Lebesgue measure: countable additivity on measurable sets, translation invariance, regularity (approximation by open/compact sets), and the existence of non-measurable sets (Vitali construction).

◆ Measurable Functions
Definition A function $f: X \to \overline{\mathbb{R}}$ is measurable (with respect to $\sigma$-algebra $\mathcal{F}$) if $\{x : f(x) > \alpha\} \in \mathcal{F}$ for every $\alpha \in \mathbb{R}$.
Egorov's Theorem If $f_n \to f$ a.e. on a set $E$ with $m(E) < \infty$, then for every $\varepsilon > 0$, there exists a measurable $A \subset E$ with $m(E \setminus A) < \varepsilon$ such that $f_n \to f$ uniformly on $A$.
Lusin's Theorem If $f$ is measurable on $E$ with $m(E) < \infty$, then for every $\varepsilon > 0$, there exists a closed set $F \subset E$ with $m(E \setminus F) < \varepsilon$ such that $f|_F$ is continuous.
05 $L^p$ Spaces

Normed spaces of integrable functions, completeness, duality, and the fundamental inequalities of Holder and Minkowski.

◆ Definition & Completeness
Definition For $1 \leq p < \infty$, the space $L^p(\mu)$ consists of measurable functions $f$ with $\|f\|_p = \left(\int |f|^p \, d\mu\right)^{1/p} < \infty$ (identifying functions equal a.e.). For $p = \infty$, $\|f\|_\infty = \text{ess sup}\,|f|$.
Riesz-Fischer Theorem $L^p(\mu)$ is a complete normed space (Banach space) for $1 \leq p \leq \infty$. In particular, $L^2(\mu)$ is a Hilbert space with inner product $\langle f, g \rangle = \int f \bar{g} \, d\mu$.
◆ Holder & Minkowski Inequalities
Holder's Inequality If $\frac{1}{p} + \frac{1}{q} = 1$ with $1 \leq p, q \leq \infty$, and $f \in L^p$, $g \in L^q$, then $fg \in L^1$ and $\|fg\|_1 \leq \|f\|_p \cdot \|g\|_q$.
Minkowski's Inequality For $1 \leq p \leq \infty$: $\|f + g\|_p \leq \|f\|_p + \|g\|_p$. This is the triangle inequality that makes $\|\cdot\|_p$ a norm.
📝 Example
Let $f(x) = x^{-1/3}$ on $(0,1)$. Determine for which $p$ we have $f \in L^p(0,1)$.
$\int_0^1 |f|^p \, dx = \int_0^1 x^{-p/3} \, dx$. This converges iff $-p/3 > -1$, i.e., $p < 3$. So $f \in L^p(0,1)$ for $1 \leq p < 3$ and $f \notin L^p(0,1)$ for $p \geq 3$.
06 Sequences of Functions

Pointwise vs uniform convergence, interchange of limits, equicontinuity, and the Arzela-Ascoli theorem.

◆ Uniform Convergence
Definition A sequence $(f_n)$ converges uniformly to $f$ on $A$ if $\sup_{x \in A} |f_n(x) - f(x)| \to 0$ as $n \to \infty$.
Weierstrass M-Test If $|f_n(x)| \leq M_n$ for all $x \in A$ and $\sum M_n < \infty$, then $\sum f_n$ converges uniformly and absolutely on $A$.

Uniform convergence preserves continuity: if $f_n \to f$ uniformly and each $f_n$ is continuous, then $f$ is continuous. Uniform convergence also allows interchange of limit and integral (on bounded intervals).

◆ Equicontinuity & Arzela-Ascoli
Definition A family $\mathcal{F}$ of functions on a metric space is equicontinuous at $x_0$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that $|f(x) - f(x_0)| < \varepsilon$ for all $f \in \mathcal{F}$ and $|x - x_0| < \delta$.
Arzela-Ascoli Theorem A subset $\mathcal{F} \subset C(K)$ (where $K$ is compact) is relatively compact (i.e., has compact closure) if and only if $\mathcal{F}$ is uniformly bounded and equicontinuous. Equivalently, every sequence in $\mathcal{F}$ has a uniformly convergent subsequence.
📝 Example
Show that $\{f_n(x) = \frac{x^n}{n}\}$ on $[0,1]$ has a uniformly convergent subsequence.
$|f_n(x)| \leq \frac{1}{n} \leq 1$ (uniformly bounded). $|f_n(x) - f_n(y)| \leq |x^n - y^n| \cdot \frac{1}{n} \leq \frac{|x - y|}{n} \leq |x - y|$ (equicontinuous with $\delta = \varepsilon$). By Arzela-Ascoli, there exists a uniformly convergent subsequence. In fact, $f_n \to 0$ uniformly since $\sup_{[0,1]} |f_n| = 1/n \to 0$.
★ Key Takeaways
✍ Practice Problems
Problem 1
Prove that $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges using the Cauchy condensation test.
Show Solution ▼
By the Cauchy condensation test, $\sum a_n$ converges iff $\sum 2^n a_{2^n}$ converges. Here $\sum 2^n \cdot \frac{1}{(2^n)^2} = \sum \frac{1}{2^n}$, which is a convergent geometric series. Hence $\sum \frac{1}{n^2}$ converges.
Problem 2
Show that $f(x) = x^2$ is uniformly continuous on $[0, a]$ for any $a > 0$, but not on $\mathbb{R}$.
Show Solution ▼
On $[0,a]$: $f$ is continuous on a compact set, so by Heine-Cantor it is uniformly continuous. On $\mathbb{R}$: take $x_n = n$, $y_n = n + 1/n$. Then $|x_n - y_n| = 1/n \to 0$ but $|f(x_n) - f(y_n)| = |2 + 1/n^2| \geq 2$. So $f$ is not uniformly continuous on $\mathbb{R}$.
Problem 3
Let $f_n(x) = n x e^{-nx^2}$ on $[0,1]$. Find $\lim_{n\to\infty} \int_0^1 f_n(x) \, dx$. Can you apply DCT?
Show Solution ▼
Pointwise, $f_n(x) \to 0$ for $x > 0$ and $f_n(0) = 0$. Direct computation: $\int_0^1 nxe^{-nx^2} dx = \frac{1}{2}(1 - e^{-n}) \to \frac{1}{2}$. But the pointwise limit is $0$, so the limit of integrals ($1/2$) differs from the integral of the limit ($0$). DCT fails here because there is no dominating $L^1$ function: $\sup_n f_n(x)$ is not integrable near $0$.
Problem 4
Construct a measurable set $E \subset [0,1]$ such that $0 < m(E \cap I) < m(I)$ for every open interval $I \subset [0,1]$.
Show Solution ▼
Use a "fat Cantor set" construction. At each stage, remove middle open intervals of decreasing length so the total removed measure is $1/2$. The remaining set $E$ has $m(E) = 1/2 > 0$, is closed and nowhere dense, and its complement $[0,1] \setminus E$ is open and dense with $m([0,1]\setminus E) = 1/2$. Both $E$ and $E^c$ intersect every open interval in positive measure.
Problem 5
Prove that if $f \in L^p \cap L^q$ with $1 \leq p < q \leq \infty$, then $f \in L^r$ for all $p \leq r \leq q$.
Show Solution ▼
Write $r = \theta p + (1-\theta)q$ for appropriate $\theta \in [0,1]$ (interpolation). Then $|f|^r = |f|^{\theta p} \cdot |f|^{(1-\theta)q}$. Apply Holder's inequality with exponents $1/\theta$ and $1/(1-\theta)$: $\|f\|_r^r \leq \|f\|_p^{\theta p} \cdot \|f\|_q^{(1-\theta)q} < \infty$. This is the $L^p$ interpolation inequality.
🎯 Test Your Understanding
1. Which of the following series converges conditionally but not absolutely?
A $\sum_{n=1}^{\infty} \frac{1}{n^2}$
B $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$
C $\sum_{n=1}^{\infty} \frac{1}{n}$
D $\sum_{n=0}^{\infty} \frac{1}{2^n}$
2. A continuous function $f:(0,1) \to \mathbb{R}$ is uniformly continuous if:
A $f$ is bounded
B $f$ can be continuously extended to $[0,1]$
C $f$ is differentiable on $(0,1)$
D $f$ is monotone
3. Let $f_n = n \chi_{(0, 1/n)}$ on $[0,1]$. Then $\int f_n \, dm = 1$ for all $n$. Which convergence theorem fails to apply?
A MCT (functions are not increasing)
B DCT (no integrable dominating function exists)
C Fatou's lemma (it gives the wrong inequality)
D All three theorems fail
4. Which of the following sets has Lebesgue measure zero?
A The irrationals in $[0,1]$
B The Cantor middle-thirds set
C $(0, 1)$
D A fat Cantor set with measure $1/4$
5. If $f \in L^2[0,1]$, which of the following is always true?
A $f \in L^1[0,1]$
B $f \in L^4[0,1]$
C $f$ is bounded a.e.
D $f$ is continuous a.e.