Understanding Limits of Sequences in Mathematics

Limits of Sequences

We will often need to specify that a number x is close to another number a. However, this doesn’t mean anything unless we specify how close. If ε is a positive number, then the statement “x is within ε of a” does have meaning. It means that the distance between x and a is less than ε – that is, |x − a| ε.

Theorem 2.1.1

If x, y, a, and ε are real numbers with ε > 0, then (a) |y|.

Theorem 2.1.2 (Triangle Inequality)

If a and b are real numbers, then (a) |a + b| ≤ |a| + |b|; and (b) ||a| − |b| ≤ |a − b|. Proof: For part (a), we observe that −|a| ≤ a ≤ |a| and −|b| ≤ b ≤ |b|. If we add these inequalities, the result is
−(|a| + |b|) ≤ a + b ≤ |a| + |b|. By the preceding theorem (with “>”).

Limits of Sequences

A sequence {a_n} converges to a number a if the distance from a_n to a can be made less than any given positive number by insisting that n be sufficiently large. More precisely:

Definition 2.1.4

A sequence {a_n} of real numbers is said to converge to the number a, or have limit equal to a, if, for each > 0, there is a real number N such that |a_n − a| ε for all n > N. In this case, we will write lim_n→∞ a_n = a or lim a_n = a or simply a_n → a.

Theorem 2.1.6

If a_n → a and a_n → b, then a = b. Proof: If a_n → a and a_n → b, then, for each > 0, there are numbers N₁ and N₂ such that n > N₁ implies |a_n − a| ε and n > N₂ implies |a_n − b| ε.