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Real Analysis

Cauchy sequence

A Cauchy sequence is a sequence whose terms become arbitrarily close to one another as the index grows large, serving as the foundational tool for defining convergence without knowing the limit.

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Definition and statement

Cauchy's Convergence Criterion for Sequences
A sequence of real numbers $\langle a_n \rangle$ is convergent in $\mathbb{R}$ if and only if it is a Cauchy sequence. That is, $\forall \varepsilon > 0, \exists N \in \mathbb{N}$ such that $|a_m - a_n| < \varepsilon$ for all $m, n \ge N$.

📐 Geometric Interpretation

Geometrically, all terms of the sequence beyond the index $N$ are crammed inside an arbitrarily small interval $(a_n - \varepsilon, a_n + \varepsilon)$ of shrinking length, illustrating that the terms cluster tightly together.

⚙️ Algebraic & Structural Meaning

It links the internal metric property of a space (mutual distance between terms) to its external topological property (existence of a limit), forming the cornerstone of completeness in $\mathbb{R}$.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition Domain / Interval Necessity What Fails If Condition Dropped
Arbitrary index separation ($m, n \ge N$) Natural numbers $N \in \mathbb{N}$ Strictly Mandatory Failing to hold for all $m, n \ge N$ allows oscillating or unbounded sequences (like $\langle n \rangle$) to mistakenly appear Cauchy.
Metric space completeness ($\mathbb{R}$ vs $\mathbb{Q}$) Underlying field $\mathbb{R}$ or $\mathbb{Q}$ Strictly Mandatory In $\mathbb{Q}$, a Cauchy sequence can converge to an irrational number (e.g., decimal approximations of $\sqrt{2}$), hence failing to converge within $\mathbb{Q}$.
Absolute difference bound ($|a_m - a_n| < \varepsilon$) Real numbers $\varepsilon > 0$ Essential for definition If distance does not become arbitrarily small, the sequence fails to stabilize.

Useful results and shortcuts

Use these identities and results when their conditions apply:

01

Sequential Ratio Test for Cauchy Property

$$|a_{n+1} - a_n| \le r^n |a_2 - a_1|, \quad 0 < r < 1$$

Rule: If the successive difference is bounded by a geometric progression with common ratio $r < 1$, the sequence is automatically Cauchy and convergent.

Application: Use this result in the stated type of problem.
02

Bounded Monotonic Sequences

$$a_n \text{ monotonic and bounded} \implies \text{Cauchy in } \mathbb{R}$$

Rule: Every monotonic bounded sequence is convergent, hence it is a Cauchy sequence. Bypass explicit $\varepsilon-N$ proofs.

Application: Directly used to resolve convergence of sequences defined by recurrence relations in DSSSB and KVS.
03

Cauchy Criterion for Series ($a_n = \sum s_n$)

$$|a_{n+p} - a_n| = |s_{n+1} + s_{n+2} + \dots + s_{n+p}| < \varepsilon$$

Rule: To test series convergence, examine tail sums directly without finding the sum to infinity.

Application: Standard pattern in UP TGT/PGT for p-series and alternating series tests.
04

Contracting Sequence Principle

$$|x_{n+2} - x_{n+1}| \le k |x_{n+1} - x_n|, \quad 0 \le k < 1$$

Rule: Immediate deduction that the sequence is Cauchy with distance bound given by geometric series sum.

Application: Appears in advanced PGT screening tests for contraction mappings.
05

Subsequence Cauchy Test

$$a_n \text{ is Cauchy} \iff \text{every subsequence is Cauchy}$$

Rule: If any single subsequence fails to be Cauchy, or if two subsequences converge to different limits, the sequence is not Cauchy.

Application: Used to eliminate oscillating sequences like $\langle (-1)^n \rangle$ instantly.
06

Rapid Elimination Trick for Unbounded Sequences

$$\lim_{n \to \infty} |a_{n+1} - a_n| \neq 0 \implies \text{Not Cauchy}$$

Rule: If the difference between consecutive terms does not approach zero, the sequence cannot be Cauchy. Instantly eliminate options.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Confusing Boundedness with Cauchy Property

Common Pitfall: Candidates assume every bounded sequence is a Cauchy sequence.

Counterexample: x_n = (-1)^n \text{ on } \mathbb{R}

Why it fails: The sequence $\langle (-1)^n \rangle$ is bounded between $-1$ and $1$, but $|x_{n+1} - x_n| = 2$ for all $n$, so it is not Cauchy and does not converge.

🛡️ Exam Defense: Boundedness is necessary for convergence in $\mathbb{R}$, but NOT sufficient. Cauchy requires terms to cluster together.
⚠️ COMMON ERROR #02

Ignoring the Underlying Space ($\mathbb{Q}$ vs $\mathbb{R}$)

Common Pitfall: Assuming all Cauchy sequences converge within the set they are defined in, forgetting field completeness.

Counterexample: x_n = \left(1 + \frac{1}{n}\right)^n \text{ or rational approximations of } \sqrt{2}

Why it fails: A sequence can be Cauchy in $\mathbb{Q}$ yet fail to converge in $\mathbb{Q}$ if its limit is irrational.

🛡️ Exam Defense: Always check if the space is complete (like $\mathbb{R}$ or $\mathbb{C}$) before deducing convergence from the Cauchy property.

Comparison

Concept / TheoremHypotheses / AssumptionsKey Formula / OutcomeGeometric MeaningExam Identification Clue
Convergent SequenceLimit exists in space$|a_n - l| < \varepsilon$Terms eventually stay inside neighborhoodLimit can be evaluated directly
Cauchy SequenceMutual distance shrinks$|a_m - n| < \varepsilon$ for $m,n \ge N$Terms cluster together arbitrarilyConsecutive difference approach zero
Bounded Sequence$m \le a_n \le M$$|a_n| \le K$Sequence stays inside a stripTerms do not shoot off to infinity
Contracting Sequence$|x_{n+1}-x_n| \le k|x_n-x_{n-1}|$Geometric boundSteps shrink by a fixed ratio $k < 1$Recursive formula with fractions

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Determine whether the sequence $x_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}$ is a Cauchy sequence in $\mathbb{R}$.

(A) Cauchy and convergent(B) Cauchy but divergent(C) Not Cauchy and divergent(D) Bounded and convergent
✓ Correct Answer: (C)
📚 Method A: Formal Textbook Derivation
Evaluate $x_{2n} - x_n = \frac{1}{n+1} + \dots + \frac{1}{2n} \ge n \cdot \frac{1}{2n} = \frac{1}{2}$. Choosing $\varepsilon = \frac{1}{3}$, no $N$ can satisfy the Cauchy condition for $m=2n, n=N$. Thus, it is not Cauchy.
⚡ Method B: 25-Second Exam Shortcut
Recognize $x_n$ as the harmonic series partial sums. Harmonic sequence is divergent, hence by Cauchy criterion, it cannot be Cauchy.
💡 Tip: Direct PYQ pattern for testing the harmonic sequence.
CANONICAL PROBLEM 02 Dual-Method Breakdown

If $0 < a < b$, and $x_1 = a, y_1 = b$, and $x_{n+1} = \sqrt{x_n y_n}$, $y_{n+1} = \frac{x_n + y_n}{2}$, then the sequence $\langle x_n \rangle$ is:

(A) Not Cauchy(B) Cauchy and converges to the arithmetic mean(C) Cauchy and converges to the geometric-arithmetic mean(D) Unbounded
✓ Correct Answer: (C)
📚 Method A: Formal Textbook Derivation
Show that $x_n \le y_n$, $\langle x_n \rangle$ is monotonically increasing, $\langle y_n \rangle$ is monotonically decreasing, and $y_{n+1} - x_{n+1} \le \frac{1}{2}(y_n - x_n)$. By nested interval property and contraction, both sequences are Cauchy and converge to the same limit.
⚡ Method B: 25-Second Exam Shortcut
AM-GM inequality preservation guarantees both sequences squeeze together rapidly, confirming Cauchy property instantly.
💡 Tip: Standard high-level PGT question requiring AM-GM convergence identification.

Practice questions

Choose an answer to check it and see the solution:

Q1
MCQ 1: Which of the following sequences is a Cauchy sequence in $\mathbb{R}$?
A $x_n = n$
B $x_n = \sin(n)$
C $x_n = \sum_{k=1}^n \frac{1}{k^2}$
D $x_n = (-1)^n n$
Derivation & Analysis:

The series $\sum \frac{1}{k^2}$ is convergent, hence its sequence of partial sums is convergent and therefore a Cauchy sequence.

Q2
MCQ 2: In the metric space $\mathbb{Q}$ with usual metric, the sequence $x_n = \left(1 + \frac{1}{n}\right)^n$ is:
A Cauchy but does not converge in $\mathbb{Q}$
B Convergent in $\mathbb{Q}$
C Not a Cauchy sequence
D Unbounded
Derivation & Analysis:

The sequence is Cauchy in $\mathbb{Q}$ (and $\mathbb{R}$), but its limit is $e$, which is irrational and hence does not belong to $\mathbb{Q}$.

Q3
MCQ 3: If a sequence $\langle a_n \rangle$ satisfies $|a_{n+1} - a_n| \le \frac{1}{2^n}$ for all $n \ge 1$, then $\langle a_n \rangle$ is:
A Divergent
B Not Cauchy
C Cauchy
D Unbounded
Derivation & Analysis:

By geometric bound on consecutive terms, the tail distance is bounded by $\sum_{k=n}^{m-1} \frac{1}{2^k} < \varepsilon$, proving it is Cauchy.

Q4
MCQ 4: Every convergent sequence in $\mathbb{R}$ is:
A Not bounded
B A Cauchy sequence
C Strictly monotonic
D Constant
Derivation & Analysis:

By definition and the completeness of $\mathbb{R}$, convergence and the Cauchy property are equivalent in $\mathbb{R}$.

Q5
MCQ 5: Which of the following is NOT true for a Cauchy sequence in $\mathbb{R}$?
A It is bounded
B It is convergent
C It has a unique limit point
D It must be monotonic
Derivation & Analysis:

A Cauchy sequence need not be monotonic (e.g., $x_n = \frac{(-1)^n}{n}$ is Cauchy and convergent, but oscillates).

Q6
MCQ 6: The sequence $x_n = 1 + \frac{1}{2!} + \frac{1}{3!} + \dots + \frac{1}{n!}$ is:
A Cauchy
B Not Cauchy
C Divergent
D Unbounded
Derivation & Analysis:

This is the sequence of partial sums for $e$, which is convergent in $\mathbb{R}$ and hence Cauchy.

Q7
MCQ 7: If $\langle a_n \rangle$ and $\langle b_n \rangle$ are two Cauchy sequences in $\mathbb{R}$, then $\langle a_n + b_n \rangle$ is:
A Never Cauchy
B Always Cauchy
C Cauchy only if $a_n = b_n$
D Convergent to 0
Derivation & Analysis:

Algebra of Cauchy sequences states that the sum, difference, and product of Cauchy sequences in $\mathbb{R}$ are also Cauchy.

Q8
MCQ 8: Assertion (A): Every Cauchy sequence of real numbers is bounded. Reason (R): Every convergent sequence in $\mathbb{R}$ is bounded.
A Both (A) and (R) are true and (R) is the correct explanation of (A)
B Both (A) and (R) are true but (R) is not the correct explanation
C (A) is true but (R) is false
D (A) is false but (R) is true
Derivation & Analysis:

Since Cauchy sequences in $\mathbb{R}$ are convergent, and convergent sequences are bounded, both statements are true and linked via completeness.

Q9
MCQ 9: The sequence $x_n = \frac{n}{n+1}$ is:
A Not Cauchy
B Cauchy
C Divergent
D Unbounded
Derivation & Analysis:

The sequence converges to $1$, hence by Cauchy criterion, it is a Cauchy sequence.

Q10
MCQ 10: If $|a_m - a_n| < \frac{1}{n} + \frac{1}{m}$ for all $m, n$, then $\langle a_n \rangle$ is:
A Not Cauchy
B Divergent
C A Cauchy sequence
D Oscillatory
Derivation & Analysis:

By choosing $N$ such that $\frac{2}{N} < \varepsilon$, the Cauchy condition is satisfied for all $m, n \ge N$.

Common questions

Convergence requires knowing the limit $l$ in advance. The Cauchy sequence criterion allows us to test whether a sequence converges using only its own internal terms, without knowing or guessing the limit beforehand.

Confusing boundedness with the Cauchy property, or failing to check whether the underlying metric space is complete (such as assuming Cauchy sequences in $\mathbb{Q}$ always converge within $\mathbb{Q}$).

Use high-yield shortcuts: check consecutive differences ($|a_{n+1} - a_n| \to 0$), examine monotonic boundedness, or apply comparison tests with known p-series and geometric series rather than constructing formal $\varepsilon-N$ inequalities.

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