Proverbs and Quotes in English
Short sentences and easy words.

• 30 approximating - √2, φ
9707#20533 SIBLINGS CHILDREN 20533

• 311 Limit (Sequence)
A sequence ｢aₙ｣ converges to a limit L if and only if for every ε>0 there exists a natural number N such that n>N ⇒ ∣aₙ−L∣<ε.
9707#9704 SIBLINGS CHILDREN COMMENT 9704

• 312 Rules (Sequence)
Suppose that ｢aₙ｣ and ｢bₙ｣ are convergent sequences with limits A and B respectively; then the following rules apply: ❶ Sum rule ｢aₙ+bₙ｣ converges to A+B. ❷ Product rule ｢aₙ·bₙ｣ converges to A·B. ❸ Quotient rule ｢aₙ/bₙ｣ converges to A/B, provided that ...
9707#3775 SIBLINGS CHILDREN COMMENT 3775

• 313 Sandwich rule (Sequence)
Let ｢aₙ｣, ｢bₙ｣ and ｢cₙ｣ be sequences satisfying aₙ≤bₙ≤cₙ for all n∈ℕ. If ｢aₙ｣ and ｢cₙ｣ both converges to the same limit L then ｢bₙ｣ also converges to L.
9707#9718 SIBLINGS CHILDREN COMMENT 9718

• 314 Composite rule (Sequence)
Let ｢aₙ｣ be a convergent sequence with limit L and let f be a continuous function whose domain contains ｢aₙ｣ and L. Then the sequence ｢f(aₙ)｣ converges to f(L).
9707#9719 SIBLINGS CHILDREN COMMENT 9719

• 321 Power rule (Sequence)
If ｢aₙ｣ is a null sequence where aₙ≥0 for all n∈ℕ, and if p∈ℝ and p>0, then ｢aₙ˄p｣ is a null sequence.
9707#9720 SIBLINGS CHILDREN COMMENT 9720

• 322 Basic null sequences
ᐥThe following are null sequences. ❶ ｢1/n˄p｣ for p>0 ❷ ｢c˄n｣ for |c|<1 ❸ ｢n˄p·c˄n｣ for p>0 and |c|<1 ❹ ｢c˄n/n!｣ for c∈ℝ ❺ ｢n˄p/n!｣ for p>0ᐥ Proof of ❶ ∀ε>0 ∃N∈ℕ such that N > 1/ε. n > N ⇒ n > 1/ε ⇒ 1/n < ε ⇒ |1/n−0| < ε ｢1/n｣ is a null sequence. By the p ...
9707#9722 SIBLINGS CHILDREN 9722

• 322_ Binomial Expansion
ᐥ(a+b)˄n = ｢｢nꞒk｣·a˄(n−k)·b˄k Σk=0,n｣ = ｢n!/((n−k)!·k!)·a˄(n−k)·b˄k Σk=0,n｣ᐥ (a+b)˄1 = ｢1!/((1−k)!·k!)·a˄(1−k)·b˄k Σk=0,1｣ = a+b (a+b)˄2 = ｢2!/((2−k)!·k!)·a˄(2−k)·b˄k Σk=0,2｣ = a˄2+2·a·b+b˄2 (a+b)˄(n+1) = (a+b)˄n·(a+b) = (a+b)˄n·a+(a+b)˄n·b = ｢n!/((n−k ...
9707#9688 SIBLINGS CHILDREN COMMENT 9688

• 335 Subsequence
ᐥThe sequence ｢a⸤n⸤r⸥⸥｣ is a subsequence of the sequence ｢a⸤n⸥｣ if ｢n⸤r⸥｣ is a strictly increasing sequence of natural numbers. ｢a⸤n⸥ Ƚn￫∞｣ = L ⇒ ｢a⸤n⸤r⸥⸥ Ƚn￫∞｣ = L ｢a⸤n⸥ Ƚn￫∞｣ = ∞ ⇒ ｢a⸤n⸤r⸥⸥ Ƚn￫∞｣ = ∞ᐥ
9707#5220 SIBLINGS CHILDREN 5220

• 341 Principle of monotone sequences
A bounded monotone sequence is convergent.
9707#9766 SIBLINGS CHILDREN 9766

• 342 Bonzano-Weierstrass theorem
A subset of Rⁿ is sequentially compact if and only if it is closed and bounded.
9707#5120 SIBLINGS CHILDREN 5120

• 342 Bonzano-Weierstrass theorem
Any bounded sequence of real numbers contains a convergent subsequence.
9707#9765 SIBLINGS CHILDREN 9765

• 343 Cauchy sequence
ᐥA sequence ｢aₙ｣ is a Cauchy sequence if and only if for every ε>0 there exists a natural number N such that n,m>N ⇒ |a⸤n⸥−a⸤m⸥|<ε. A sequence is convergent if and only if the sequence is a Cauchy sequence.ᐥ If ｢a⸤n⸥｣ is a convergent sequence, then for a ...
9707#5221 SIBLINGS CHILDREN 5221

• 34_ ｢(1+1/n)˄n｣ is convergent
(1+1/n)˄n = ｢n!/((n−i)!·i!)·1˄(n−i)·(1/n)˄i Σi=0,n｣ = ｢(1/i!)·n!/(n−i)!·(1/n˄i) Σi=0,n｣ = ｢(1/i!)·｢(n−j) Πj=0,i−1｣·(1/n˄i) Σi=0,n｣ = ｢(1/i!)·｢(n−j)/n Πj=0,i−1｣ Σi=0,n｣ = ｢(1/i!)·｢(1−j/n) Πj=0,i−1｣ Σi=0,n｣ < ｢(1/i!)·｢(1−j/(n+1)) Πj=0,i−1｣ Σi=0,n｣ < ｢(1/i!) ...
9707#9767 SIBLINGS CHILDREN 9767

• harmonic numbers
9707#8621 SIBLINGS CHILDREN 8621