Sequences and Limits

Jiří Lebl · Basic Analysis I, Ch. 2 · CC BY-SA 4.0

A sequence converges to L if, for every ε > 0, all but finitely many terms lie within ε of L. Completeness guarantees that every bounded monotone sequence converges and every Cauchy sequence converges.

Convergence: the epsilon-N definition

A sequence (a_n) converges to L if for every ε > 0, there exists N such that for all n ≥ N, |a_n - L| < ε. The key: you pick any ε, no matter how small, and I produce an N that works.

Scheme

; Sequence a_n = 1/n converges to 0
; For any epsilon, find N such that 1/n < epsilon for all n >= N

(define (find-N epsilon)
  (let loop ((n 1))
    (if (< (/ 1.0 n) epsilon) n (loop (+ n 1)))))

(display "epsilon = 0.1, N = ")
(display (find-N 0.1)) (newline)
(display "epsilon = 0.01, N = ")
(display (find-N 0.01)) (newline)
(display "epsilon = 0.001, N = ")
(display (find-N 0.001)) (newline)

; Verify: all terms past N are within epsilon
(define (verify-convergence epsilon)
  (let* ((N (find-N epsilon))
         (checks (list N (+ N 1) (+ N 5) (+ N 50))))
    (for-each (lambda (n)
      (display "  a_")
      (display n)
      (display " = ")
      (display (/ 1.0 n))
      (display (if (< (/ 1.0 n) epsilon) " < epsilon" " >= epsilon"))
      (newline))
    checks)))

(display "Checking epsilon = 0.05:") (newline)
(verify-convergence 0.05)

Limit theorems

Limits respect arithmetic: if a_n → a and b_n → b, then a_n + b_n → a + b, a_n * b_n → a * b, and (if b ≠ 0) a_n / b_n → a / b. The squeeze theorem: if a_n ≤ c_n ≤ b_n and both a_n and b_n converge to L, then c_n → L.

Scheme

; Limit theorems: sum, product, squeeze
; a_n = 1 + 1/n -> 1
; b_n = 2 - 1/n -> 2

(define (a n) (+ 1 (/ 1.0 n)))
(define (b n) (- 2 (/ 1.0 n)))

(display "Sum: a_n + b_n -> 1 + 2 = 3") (newline)
(display "  n=100: ")
(display (+ (a 100) (b 100))) (newline)

(display "Product: a_n * b_n -> 1 * 2 = 2") (newline)
(display "  n=100: ")
(display (* (a 100) (b 100))) (newline)

; Squeeze: -1/n <= sin(n)/n <= 1/n, both bounds -> 0
(display "Squeeze: |sin(n)/n| <= 1/n -> 0") (newline)
(display "  sin(100)/100 = ")
(display (/ (sin 100) 100)) (newline)
(display "  1/100         = ")
(display (/ 1.0 100))

Monotone convergence theorem

Every bounded, monotone sequence converges. If a sequence is increasing and bounded above, it converges to its supremum. This is a direct consequence of the completeness axiom.

Scheme

; Monotone convergence: a_n = sum_{k=1}^{n} 1/k!
; Increasing and bounded above by 3, converges to e

(define (factorial n)
  (if (<= n 1) 1 (* n (factorial (- n 1)))))

(define (partial-e n)
  (let loop ((k 0) (sum 0.0))
    (if (> k n) sum
      (loop (+ k 1) (+ sum (/ 1.0 (factorial k)))))))

(display "Partial sums of 1/k! (converging to e):") (newline)
(for-each (lambda (n)
  (display "  n=")
  (display n)
  (display ": ")
  (display (partial-e n))
  (newline))
  (list 1 2 5 10 15))

(display "Bounded above by 3? ")
(display (< (partial-e 15) 3))

Bolzano-Weierstrass theorem

Every bounded sequence has a convergent subsequence. This is the Bolzano-Weierstrass theorem, the key compactness result for sequences. Even if a sequence doesn't converge (like (-1)^n), some subsequence does.

Scheme

; Bolzano-Weierstrass: bounded sequence has convergent subsequence
; a_n = (-1)^n * (1 + 1/n) does not converge
; but the subsequence a_{2k} = 1 + 1/(2k) -> 1

(define (a n)
  (* (if (even? n) 1 -1)
     (+ 1 (/ 1.0 n))))

(display "Full sequence (diverges):") (newline)
(for-each (lambda (n)
  (display "  a_")
  (display n)
  (display " = ")
  (display (a n))
  (newline))
  (list 1 2 3 4 5 6 7 8))

(display "Even subsequence (converges to 1):") (newline)
(for-each (lambda (k)
  (let ((n (* 2 k)))
    (display "  a_")
    (display n)
    (display " = ")
    (display (a n))
    (newline)))
  (list 1 2 5 10 50 100))

Cauchy sequences

A sequence is Cauchy if its terms get arbitrarily close to each other: for every ε, there exists N such that |a_m - a_n| < ε for all m, n ≥ N. In the reals, a sequence converges if and only if it is Cauchy. This equivalence is another way to state completeness.

Scheme

; Cauchy criterion: terms get close to each other
; a_n = sum_{k=1}^{n} 1/k^2 is Cauchy (converges to pi^2/6)

(define (partial-sum n)
  (let loop ((k 1) (s 0.0))
    (if (> k n) s
      (loop (+ k 1) (+ s (/ 1.0 (* k k)))))))

; Check Cauchy: |a_m - a_n| for large m, n
(display "Cauchy check for sum 1/k^2:") (newline)
(let* ((a100 (partial-sum 100))
       (a200 (partial-sum 200))
       (a500 (partial-sum 500))
       (a1000 (partial-sum 1000)))
  (display "  |a_200 - a_100| = ")
  (display (abs (- a200 a100))) (newline)
  (display "  |a_500 - a_200| = ")
  (display (abs (- a500 a200))) (newline)
  (display "  |a_1000 - a_500| = ")
  (display (abs (- a1000 a500))) (newline)
  (display "  a_1000 = ")
  (display a1000) (newline)
  (display "  pi^2/6 = ")
  (display (/ (* 3.141592653589793 3.141592653589793) 6.0)))

Notation reference

Symbol	Scheme	Meaning
lim a_n = L	(find-N epsilon)	Convergence: terms approach L
\|a_n - L\| < ε	(< (abs (- an L)) eps)	Within epsilon of the limit
Cauchy	\|a_m - a_n\| < ε	Terms get close to each other
BW theorem	subsequence	Bounded seq has convergent subseq

Neighbors

Ch. 1: Real Numbers — completeness makes convergence work
Grinstead Ch. 8: Law of Large Numbers — convergence of random variables uses these same definitions
🎰 Grinstead Ch.9 CLT — the Central Limit Theorem as a convergence result
🔢 Discrete Math Ch.2 — sequences and recurrences: the combinatorial counterpart to analytic limits
📐 Calculus Ch.3 — informal limits in calculus made rigorous here

Translation notes

We compute convergence numerically but the real content is the proofs. The epsilon-N definition is an existence statement: "there exists N such that..." Our code finds a specific N, which is stronger than existence but weaker than a proof that it always works.

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