Fundamental theorems on sequences
Theorem 2.2.1 (Uniqueness of the limit). If {an} is a convergent sequence with limit ℓ, there are no other limits.
Proof. Suppose there exist two limits ℓ1 and ℓ2 for the same sequence. By using triangular inequality:
|ℓ1 − ℓ2| = |ℓ1 − an + an − ℓ2| ≤ |ℓ1 − an| + |an − ℓ2| < 2ε
Since ε is a real positive number (can be chosen to be arbitrarily small), this inequality can only hold if ℓ1 = ℓ2. □
Theorem 2.2.2. (Squeeze Theorem). Let {an}, {bn} and {cn} be sequences such that
an ≤ bn ≤ cn
for every positive integer n. If
limn ⟶ ∞ an = ℓ = limn ⟶ ∞ cn
then
limn ⟶ ∞ bn = ℓ
Proof. Let ε > 0. By the definition of limit, there exists a positive integer N1 such that if n ≥ N1:
ℓ − ε < an < ℓ
An there exists a positive integer N2 such that if n ≥ N2:
ℓ − ε < cn < ℓ + ε
Thus is n ≥ max {N1,N2}, we have
ℓ − ε < an ≤ bn ≤ cn < ℓ + ε □
Example 2.2.3. To illustrate the squeeze theorem, we find the limit of the following sequence (sin n)/n as n → ∞. Recall that the sine function has range [−1, +1] so
−1 ≤ sin n ≤ 1
for every value of n; when we divide these inequalities by n, we obtain
−1/n ≤ (sin n)/n ≤ 1/n
The flanking sequences are −1∕n and 1∕n, and both of them converge to 0 as n → ∞. Therefore, {an} converges to 0 too. ■
Theorem 2.2.4. (Constancy of sign) If a sequence {an} has limit ℓ ≠ 0, its terms starting from an index ν on, have the same sign of ℓ.
Proof. Let ε= ℓ/2. By the definition of limit:
ℓ − |ℓ|/2 < an < ℓ + |ℓ|/2
∀ n > ν, given a certain ν (whose value depends on the choice of ε). Thus if ℓ > 0 it follows (from the first inequality)
an > ℓ/2 > 0
or if ℓ < 0 it follows (from the second inequality)
an < ℓ/2 < 0. □