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 power rule, ｢(1/n)˄p｣ is also a null sequence.
Proof of ❷
(1) c>0
∀c in (0,1) ∃x>0 such that c=1/(1+x).
∃N∈ℕ such that n>N ⇒
|cⁿ|
= |1/(1+x)ⁿ|
┄
Binomial expansion
(1+x)ⁿ
= ｢｢nꞒk｣·x˄k Σk=0,n｣
≥ ｢｢nꞒk｣·x˄k Σk=0,1｣
= 1+nx
┄
≤ |1/(1+nx)|
< |1/(nx)|
= |(1/x)·(1/n)|
┄
｢1/n｣ is a null sequence and so is ｢(1/x)·(1/n)｣ by the scalar product rule.
┄
< ε
(2) c=0
｢0ⁿ｣ is a null sequence.
(3) c<0
∀ε>0 ∃N∈ℕ such that n>N ⇒ |cⁿ| = |-cⁿ| < ε.
Proof of ❸
(1) p=1, 0<c, n≥2
∀c in (0,1) ∃x>0 such that c=1/(1+x).
∃N∈ℕ such that n>N ⇒
|n·cⁿ|
= |n/(1+x)ⁿ|
┄
Binomial expansion
(1+x)ⁿ
= ｢｢nꞒk｣·x˄k Σk=0,n｣
≥ ｢｢nꞒk｣·x˄k Σk=2,2｣
= n!/((n−2)!·2!)·x²
= n(n−1)x²/2
┄
≤ |n·2/(n(n−1)x²)|
= |(2/x²)·(1/(n−1))|
┄
｢1/n｣ is a null sequence and so is ｢(1/x)·(1/n)｣ by the scalar product rule.
┄
< ε
(2) c=0
｢n·0ⁿ｣ is a null sequence.
(3) c<0
∀ε>0 ∃N∈ℕ such that n>N ⇒ |n·cⁿ| = |-n·cⁿ| < ε.
(4) p>0
|c|<1 ⇒ |c˄(1/p)|<1
｢n·(c˄(1/p))˄n｣ is a null sequence for p>0 and |c|<1.
By the power rule, ｢(n·(c˄(1/p))˄n)˄p｣=｢n˄p·c˄n｣ is a null sequence for p>0 and |c|<1.
Proof of ❹
(1) c>0
∃m∈ℤ such that m+1>c.
∀ε>0 ∃N>m in ℕ such that n>N ⇒
|c˄n/n!|
= |｢c/k Ꝑk=1,n｣|
= |｢c/k Ꝑk=1,m｣·｢c/k Ꝑk=m+1,n｣|
┄
c/k < 1 for k ≥ m+1
｢c/k Ꝑk=m+1,n｣ ≤ c/n for k ≥ m+1
┄
≤ |｢c/k Ꝑk=1,m｣·(c/n)|
┄
｢1/n｣ is a null sequence and so is ｢c/k Ꝑk=1,m｣·c·(1/n) by the scalar product rule.
┄
< ε
(2) c=0
｢0ⁿ/n!｣ is a null sequence.
(3) c<0
∀ε>0 ∃N∈ℕ such that n>N ⇒ |cⁿ/n!| = |-cⁿ/n!| < ε.
Proof of ❺
｢n˄p/n!｣
= ｢n˄p·(1/2)˄n·2˄n/n!｣
┄
｢n˄p·c˄n｣ is a null sequence for |c|<1 and so is ｢n˄p·(1/2)˄n｣.
｢c˄n/n!｣ is a null sequence for c∈ℝ and so is ｢2˄n/n!｣.
┄
By the product rule, ｢n˄p/n!｣ is a null sequence.
