Real Analysis

Master sequences, series, continuity, differentiability, Riemann integration, and metric spaces for GATE Mathematics.

Sequences & Series Continuity & Differentiability Riemann Integration Function Sequences Metric Spaces

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Sequences and Series

Convergence of real sequences and infinite series forms the backbone of analysis. Understanding these concepts is essential for all subsequent topics.

◆ Convergence of Sequences

Definition — Convergence A sequence $(a_n)$ 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 $n \geq N$. We write $\lim_{n \to \infty} a_n = L$.

Theorem — Monotone Convergence Every bounded monotone sequence in $\mathbb{R}$ converges. Specifically, a monotone increasing sequence bounded above converges to its supremum, and a monotone decreasing sequence bounded below converges to its infimum.

A Cauchy sequence is one where $|a_m - a_n| \to 0$ as $m, n \to \infty$; in $\mathbb{R}$, Cauchy $\iff$ convergent
Bolzano-Weierstrass: Every bounded sequence in $\mathbb{R}$ has a convergent subsequence
$\limsup$ and $\liminf$ always exist for bounded sequences

◆ Convergence of Series

An infinite series $\sum_{n=1}^{\infty} a_n$ converges if the sequence of partial sums $S_N = \sum_{n=1}^{N} a_n$ converges.

Comparison test: If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges
Ratio test: If $L = \lim |a_{n+1}/a_n|$, then the series converges if $L < 1$ and diverges if $L > 1$
Root test: If $L = \limsup |a_n|^{1/n}$, then the series converges if $L < 1$ and diverges if $L > 1$
Leibniz test: An alternating series $\sum (-1)^n b_n$ converges if $(b_n)$ is decreasing and $b_n \to 0$
Absolute convergence implies convergence, but not conversely

★ Example

Prove that $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges using the comparison test.

Step 1: For $n \geq 2$, observe that $\dfrac{1}{n^2} \leq \dfrac{1}{n(n-1)} = \dfrac{1}{n-1} - \dfrac{1}{n}$ (partial fractions).

Step 2: The partial sums telescope: $\displaystyle\sum_{n=2}^{N} \left(\frac{1}{n-1} - \frac{1}{n}\right) = 1 - \frac{1}{N} \to 1$ as $N \to \infty$.

Step 3: By comparison, $\displaystyle\sum_{n=2}^{\infty} \frac{1}{n^2} \leq 1$, so $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \sum_{n=2}^{\infty} \frac{1}{n^2} \leq 2$.

Since the partial sums are bounded and increasing (all terms are positive), the series converges.

★ Key Takeaways — Sequences & Series

The harmonic series $\sum 1/n$ diverges, but $\sum 1/n^p$ converges for $p > 1$ (p-series test)
The ratio and root tests are inconclusive when the limit equals 1
Absolute convergence is strictly stronger than conditional convergence

Continuity and Differentiability

The precise definitions of continuity and differentiability, along with the powerful Mean Value Theorems that connect them.

◆ Continuity

Definition — Epsilon-Delta Continuity A function $f: D \to \mathbb{R}$ is continuous at $c \in D$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $|x - c| < \delta$ implies $|f(x) - f(c)| < \varepsilon$.

Uniform continuity: $\delta$ depends only on $\varepsilon$, not on the point $c$
Every continuous function on a closed bounded interval $[a,b]$ is uniformly continuous (Heine-Cantor)
Intermediate Value Theorem: If $f$ is continuous on $[a,b]$ and $f(a) < k < f(b)$, then $f(c) = k$ for some $c \in (a,b)$
Extreme Value Theorem: A continuous function on $[a,b]$ attains its maximum and minimum

◆ Differentiability and Mean Value Theorems

$f$ is differentiable at $c$ if $\lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$ exists. Differentiability implies continuity, but not conversely (e.g., $f(x) = |x|$ at $x = 0$).

Theorem — Mean Value Theorems Rolle's Theorem: If $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then $f'(c) = 0$ for some $c \in (a,b)$.

Lagrange's MVT: Under the same conditions (without $f(a)=f(b)$), there exists $c \in (a,b)$ with $f'(c) = \frac{f(b)-f(a)}{b-a}$.

Cauchy's MVT: If $f, g$ are continuous on $[a,b]$, differentiable on $(a,b)$, and $g'(x) \neq 0$, then $\frac{f'(c)}{g'(c)} = \frac{f(b)-f(a)}{g(b)-g(a)}$ for some $c \in (a,b)$.

★ Example

Use the Mean Value Theorem to show that $|\sin x - \sin y| \leq |x - y|$ for all $x, y \in \mathbb{R}$.

Step 1: Let $f(t) = \sin t$. Then $f$ is continuous and differentiable everywhere, with $f'(t) = \cos t$.

Step 2: By Lagrange's MVT, for $x \neq y$, there exists $c$ between $x$ and $y$ such that $\frac{\sin x - \sin y}{x - y} = \cos c$.

Step 3: Since $|\cos c| \leq 1$, we get $|\sin x - \sin y| = |\cos c| \cdot |x - y| \leq |x - y|$.

This also proves that $\sin$ is Lipschitz continuous (and hence uniformly continuous) on $\mathbb{R}$.

Taylor's theorem: If $f$ has $n+1$ continuous derivatives on $[a,b]$, then $f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k + \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}$ for some $c$ between $a$ and $x$ (Lagrange remainder form).

★ Key Takeaways — Continuity & Differentiability

Differentiable $\Rightarrow$ continuous, but continuous $\not\Rightarrow$ differentiable
Uniform continuity is a global property; standard continuity is local
Cauchy's MVT is the key to proving L'Hopital's rule

Riemann Integration

The rigorous construction of the integral via partitions and upper/lower sums, culminating in the Fundamental Theorem of Calculus.

◆ Riemann Sums and Integrability

Definition — Riemann Integrability For a bounded function $f$ on $[a,b]$, the upper sum $U(f,P) = \sum M_i \Delta x_i$ and lower sum $L(f,P) = \sum m_i \Delta x_i$ where $M_i, m_i$ are the supremum and infimum of $f$ on each subinterval. $f$ is Riemann integrable if $\inf_P U(f,P) = \sup_P L(f,P)$.

Every continuous function on $[a,b]$ is Riemann integrable
Every monotone function on $[a,b]$ is Riemann integrable
A bounded function is Riemann integrable if and only if its set of discontinuities has measure zero (Lebesgue's criterion)

◆ Fundamental Theorem and Improper Integrals

Theorem — Fundamental Theorem of Calculus Part 1: If $f$ is integrable on $[a,b]$ and $F(x) = \int_a^x f(t)\,dt$, then $F$ is continuous. If $f$ is continuous at $c$, then $F'(c) = f(c)$.

Part 2: If $f$ is integrable on $[a,b]$ and $F$ is an antiderivative of $f$ (i.e., $F' = f$), then $\int_a^b f(x)\,dx = F(b) - F(a)$.

★ Example

Determine whether $\displaystyle\int_1^{\infty} \frac{1}{x^p}\,dx$ converges, and for which values of $p$.

Case $p \neq 1$: $\displaystyle\int_1^{R} x^{-p}\,dx = \left[\frac{x^{1-p}}{1-p}\right]_1^R = \frac{R^{1-p} - 1}{1-p}$.

As $R \to \infty$: if $p > 1$, then $1 - p < 0$, so $R^{1-p} \to 0$ and the integral equals $\frac{1}{p-1}$ (converges).

If $p < 1$, then $1 - p > 0$, so $R^{1-p} \to \infty$ (diverges).

Case $p = 1$: $\displaystyle\int_1^{R} \frac{1}{x}\,dx = \ln R \to \infty$ (diverges).

Conclusion: The integral converges if and only if $p > 1$.

★ Key Takeaways — Riemann Integration

The integral $\int_1^\infty x^{-p}\,dx$ converges iff $p > 1$ (compare with the p-series)
FTC connects differentiation and integration; the antiderivative need not be elementary
For improper integrals, always check convergence before evaluating

Sequences and Series of Functions

Understanding when limits and infinite sums of functions preserve continuity, differentiability, and integrability.

◆ Pointwise vs Uniform Convergence

Definition — Uniform Convergence A sequence of functions $(f_n)$ converges uniformly to $f$ on $S$ if $\sup_{x \in S} |f_n(x) - f(x)| \to 0$ as $n \to \infty$. Equivalently, for every $\varepsilon > 0$ there exists $N$ (independent of $x$) such that $|f_n(x) - f(x)| < \varepsilon$ for all $n \geq N$ and all $x \in S$.

Pointwise convergence: $f_n(x) \to f(x)$ for each fixed $x$ (the $N$ may depend on $x$)
Uniform limit of continuous functions is continuous (fails for pointwise limits)
Under uniform convergence, $\lim \int f_n = \int \lim f_n$

Theorem — Weierstrass M-test If $|f_n(x)| \leq M_n$ for all $x \in S$ and $\sum M_n$ converges, then $\sum f_n$ converges uniformly and absolutely on $S$.

★ Example

Show that $f_n(x) = x^n$ converges pointwise but not uniformly on $[0,1)$.

Pointwise: For each $x \in [0,1)$, $|x| < 1$, so $x^n \to 0$. Thus $f_n \to 0$ pointwise.

Not uniform: $\sup_{x \in [0,1)} |x^n - 0| = \sup_{x \in [0,1)} x^n = 1$ (the supremum approaches 1 as $x \to 1^-$). Since $\sup |f_n(x)| = 1 \not\to 0$, convergence is not uniform.

◆ Power Series

A power series $\sum_{n=0}^{\infty} a_n (x - c)^n$ has a radius of convergence $R$ given by $\frac{1}{R} = \limsup_{n \to \infty} |a_n|^{1/n}$. The series converges absolutely for $|x-c| < R$ and diverges for $|x-c| > R$.

Power series converge uniformly on any compact subset of the interval of convergence
Power series can be differentiated and integrated term by term within the interval of convergence
The endpoints $x = c \pm R$ must be checked separately

★ Key Takeaways — Function Sequences

Uniform convergence preserves continuity; pointwise convergence does not
The Weierstrass M-test is the primary tool for showing uniform convergence of series
Radius of convergence: use the root test formula $1/R = \limsup |a_n|^{1/n}$

Metric Spaces

Generalizing the concepts of convergence, continuity, and completeness from $\mathbb{R}$ to abstract metric spaces.

◆ Topology of Metric Spaces

Definition — Metric Space A metric space $(X, d)$ is a set $X$ with a function $d: X \times X \to [0, \infty)$ satisfying: (i) $d(x,y) = 0 \iff x = y$, (ii) $d(x,y) = d(y,x)$, (iii) $d(x,z) \leq d(x,y) + d(y,z)$ (triangle inequality).

A set $U$ is open if every point has an open ball contained in $U$
A set $F$ is closed if its complement is open, equivalently if it contains all its limit points
A metric space is complete if every Cauchy sequence converges (e.g., $\mathbb{R}$ is complete, $\mathbb{Q}$ is not)
A set is compact if every open cover has a finite subcover; in $\mathbb{R}^n$, compact $\iff$ closed and bounded (Heine-Borel)

◆ Contraction Mapping and Connectedness

Theorem — Banach Fixed Point (Contraction Mapping) Let $(X, d)$ be a complete metric space and $T: X \to X$ a contraction (i.e., $d(Tx, Ty) \leq \alpha\, d(x,y)$ for some $0 \leq \alpha < 1$). Then $T$ has a unique fixed point $x^* \in X$, and for any $x_0 \in X$, the sequence $x_{n+1} = T(x_n)$ converges to $x^*$.

★ Example

Show that $T(x) = \frac{1}{2}(x + 3/x)$ has a unique fixed point on $[1, 3]$ with the standard metric.

Step 1: $T$ maps $[1,3]$ into itself: for $x \in [1,3]$, $T(x) = \frac{x}{2} + \frac{3}{2x}$. At $x=1$: $T(1) = 2$; at $x=3$: $T(3) = 2$. By calculus, $T'(x) = \frac{1}{2} - \frac{3}{2x^2}$, and $T$ maps $[1,3] \to [\sqrt{3}, 2] \subset [1,3]$.

Step 2: $|T'(x)| = \left|\frac{1}{2} - \frac{3}{2x^2}\right| \leq \frac{1}{2}$ for $x \in [1,3]$. By the Mean Value Theorem, $|T(x) - T(y)| \leq \frac{1}{2}|x - y|$.

Step 3: Since $[1,3]$ is a complete metric space and $T$ is a contraction with $\alpha = 1/2$, the Banach fixed point theorem guarantees a unique fixed point. Solving $x = \frac{1}{2}(x + 3/x)$ gives $x = \sqrt{3}$.

A metric space is connected if it cannot be written as a union of two disjoint nonempty open sets. Subsets of $\mathbb{R}$ are connected if and only if they are intervals.

★ Key Takeaways — Metric Spaces

Completeness + contraction = unique fixed point (Banach's theorem is a GATE favourite)
In $\mathbb{R}^n$: compact $\iff$ closed and bounded (Heine-Borel)
Every compact metric space is complete, but not conversely

✎ Practice Problems

Problem 1

Determine whether the sequence $a_n = \left(1 + \frac{1}{n}\right)^n$ converges, and if so, find its limit.

Show Solution ▼

The sequence is monotone increasing and bounded above by 3. By the monotone convergence theorem, it converges. The limit is $e = 2.71828\ldots$. This is in fact the definition of Euler's number $e$.

Problem 2

Does $\displaystyle\sum_{n=1}^{\infty} \frac{n!}{n^n}$ converge or diverge?

Show Solution ▼

Apply the ratio test: $\frac{a_{n+1}}{a_n} = \frac{(n+1)! \cdot n^n}{(n+1)^{n+1} \cdot n!} = \frac{n^n}{(n+1)^n} = \left(\frac{n}{n+1}\right)^n = \frac{1}{(1+1/n)^n} \to \frac{1}{e} < 1$. The series converges.

Problem 3

Let $f(x) = x^2 \sin(1/x)$ for $x \neq 0$ and $f(0) = 0$. Is $f$ differentiable at $x = 0$?

Show Solution ▼

$f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{h^2 \sin(1/h)}{h} = \lim_{h \to 0} h \sin(1/h) = 0$ (by squeeze theorem since $|h\sin(1/h)| \leq |h|$). So $f$ is differentiable at $0$ with $f'(0) = 0$.

Problem 4

Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{2^n + 1}$.

Show Solution ▼

$a_n = \frac{1}{2^n + 1}$. Then $|a_n|^{1/n} = \frac{1}{(2^n+1)^{1/n}} \to \frac{1}{2}$ since $(2^n+1)^{1/n} \to 2$. By the root test formula, $R = 1/\limsup|a_n|^{1/n} = 1/(1/2) = 2$. The radius of convergence is $R = 2$.

Problem 5

Show that the set $A = \{1/n : n \in \mathbb{N}\}$ is not compact in $\mathbb{R}$.

Show Solution ▼

$A$ is bounded but not closed (the limit point $0$ is not in $A$: $1/n \to 0$ but $0 \notin A$). By Heine-Borel, a subset of $\mathbb{R}$ is compact iff closed and bounded. Since $A$ is not closed, it is not compact. Alternatively, the open cover $\{(1/(n+1), 1/(n-1)) : n \geq 2\} \cup \{(1/2, 2)\}$ covers $A$ but has no finite subcover.

⚙ Interactive Quiz

1. The series $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$ is:

A Absolutely convergent

B Conditionally convergent

C Divergent

D Oscillating

2. Which function is uniformly continuous on $(0, 1)$?

A $f(x) = 1/x$

B $f(x) = \sin(1/x)$

C $f(x) = \sqrt{x}$

D $f(x) = e^{1/x}$

3. If $f$ is Riemann integrable on $[0,1]$, which of the following is necessarily true?

A $f$ is continuous on $[0,1]$

B $f$ is bounded on $[0,1]$

C $f$ is differentiable a.e.

D $f$ is monotone on $[0,1]$

4. The sequence $f_n(x) = nx e^{-nx^2}$ on $[0, 1]$:

A Converges uniformly to 0

B Converges pointwise to 0 but not uniformly

C Diverges pointwise

D Converges uniformly to $e^{-x}$

5. Which of the following metric spaces is complete?

A $(\mathbb{Q}, |\cdot|)$

B $((0,1), |\cdot|)$

C $([0,1], |\cdot|)$

D $(\mathbb{R} \setminus \mathbb{Q}, |\cdot|)$