Fundamental theorems on sequences

Theorem 2.1 (Uniqueness of the limit). If {a_n} 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 − a_n + a_n − ℓ₂| ≤ |ℓ₁ − a_n| + |a_n − 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 - (Comparison test). Let {a_n}, {b_n} and {c_n} three convergent sequences with limit ℓ. If

a_n ≤ b_n ≤ c_n, a_n → ℓ, c_n → ℓ ⇒ b_n → ℓ

Proof: Let ε > 0. By the definition of limit:

ℓ - ε < a_n < ℓ + ε; ℓ - ε < c_n < ℓ + ε

it follows that

ℓ - ε < a_n ≤ b_n ≤ c_n < ℓ + ε

and definetly

ℓ - ε < b_n < ℓ + ε

Thus b_n → ℓ.□

Theorem 2.3 - (permanence of sing). If a sequence {a_n} 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 < a_n < 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)

a_n > ℓ/2 > 0

or if ℓ < 0 it follows (from the second inequality)

a_n < ℓ/2 < 0. □