Sequences

A sequence in \( \mathbb{R} \) is a function \( x: \mathbb{N} \to \mathbb{R} \), written as \( (x_n)_{n=1}^{\infty} \) or simply \( (x_n) \). Each \( x_n \) is called a term of the sequence.

Hinglish: Sequence = ek infinite list jo ek rule follow karti hai. Natural numbers (1, 2, 3, ...) ko input maano, aur function output deta hai:

• \( x_n = 1/n \) → Terms: \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots \) (chhote hote jaate hain, 0 ki taraf)
• \( x_n = (-1)^n \) → Terms: \( -1, 1, -1, 1, \dots \) (oscillate karta hai)
• \( x_n = n^2 \) → Terms: \( 1, 4, 9, 16, \dots \) (infinity ki taraf jaata hai)

A sequence \( (x_n) \) is bounded if there exists \( M > 0 \) such that \( |x_n| \leq M \) for all \( n \). It stays within \( [-M, M] \) forever.

Hinglish: Bounded sequence = saare terms ek dabba \( [-M, M] \) ke andar rehte hain, kabhi bahar nahi jaate. Example: \( x_n = \sin(n) \) bounded hai (\( M = 1 \) se kaam chal jayega, kyunki \( -1 \leq \sin(n) \leq 1 \) hamesha). \( x_n = n \) bounded nahi hai (1, 2, 3, ... koi M nahi rok sakta isko).

A sequence is monotone increasing if \( x_n \leq x_{n+1} \) for all \( n \) (never goes down). It is monotone decreasing if \( x_n \geq x_{n+1} \) for all \( n \) (never goes up).

Hinglish: Monotone = hamesha ek hi direction mein chalti hai, kabhi peeche nahi mudti. Example: \( 1, 2, 3, 4, \dots \) monotone increasing ✅. \( 1, 1/2, 1/3, 1/4, \dots \) monotone decreasing ✅. \( 1, -1, 1, -1, \dots \) monotone nahi ❌ (baar baar direction badal rahi hai).

We say \( x_n \to x \) if for every \( \varepsilon > 0 \) there exists \( N \in \mathbb{N} \) such that \( n \geq N \implies |x_n - x| < \varepsilon \).

Hinglish: Socho ek classroom (bounded box) mein 100 students hain jo random jagah baithe hain. BW theorem kehta hai ki tum hamesha ek line (subsequence) bana sakte ho students ki jo kisi ek point ki taraf converge karegi — chahe baithne ka pattern kitna bhi random ho!

Key consequence: This theorem is the backbone of compactness. A set is compact \( \iff \) every sequence in it has a convergent subsequence (sequential compactness).

A sequence \( (x_n) \) is Cauchy if for every \( \varepsilon > 0 \), there exists \( N \in \mathbb{N} \) such that \( m, n \geq N \implies |x_m - x_n| < \varepsilon \).

Hinglish: Cauchy sequence mein hum limit ke baare mein kuch nahi kehte — sirf yeh dekhte hain ki terms aapas mein paas aa rahe hain ya nahi. Socho: agar exam mein tumhare last 10 tests ke marks 95, 96, 95.5, 96.1, 95.8, ... aa rahe hain — toh ye Cauchy hai (marks ek doosre ke paas aa rahe hain), bhale hi tumhe nahi pata exact limit kya hoga.