Introduction to Mathematical Analysis I - Second Edition

65 Corollary 3.1.3 Let f : D → R and let ¯ x be a limit point of D . If f has a limit at ¯ x , then this limit is unique. Proof: Suppose by contradiction that f has two different limits ` 1 and ` 2 . Let { x n } be a sequence in D \ { ¯ x } that converges to ¯ x . By Theorem 3.1.2 , the sequence { f ( x n ) } converges to two different limits ` 1 and ` 2 . This is a contradiction to Theorem 2.1.3 . The following corollary follows directly from Theorem 3.1.2 . Corollary 3.1.4 Let f : D → R and let ¯ x be a limit point of D . Then f does not have a limit at ¯ x if and only if there exists a sequence { x n } in D such that x n 6 = ¯ x for every n , { x n } converges to ¯ x , and { f ( x n ) } does not converge. Example 3.1.5 Consider the Dirichlet function f : R → R given by f ( x ) = ( 1 , if x ∈ Q ; 0 , if x ∈ Q c . Then lim x → ¯ x f ( x ) does not exist for any ¯ x ∈ R . Indeed, fix ¯ x ∈ R and choose two sequences { r n } , { s n } converging to ¯ x such that r n ∈ Q and s n 6∈ Q for all n ∈ N . Define a new sequence { x n } by x n = ( r k , if n = 2 k ; s k , if n = 2 k − 1 . It is clear that { x n } converges to ¯ x . Moreover, since { f ( r n ) } converges to 1 and { f ( s n ) } converges to 0, Theorem 2.1.9 implies that the sequence { f ( x n ) } does not converge. It follows from the sequential characterization of limits that lim x → ¯ x f ( x ) does not exist. Theorem 3.1.5 Let f , g : D → R and let ¯ x be a limit point of D . Suppose that lim x → ¯ x f ( x ) = ` 1 , lim x → ¯ x g ( x ) = ` 2 , and that there exists δ > 0 such that f ( x ) ≤ g ( x ) for all x ∈ B ( ¯ x ; δ ) ∩ D , x 6 = ¯ x . Then ` 1 ≤ ` 2 . Proof: Let { x n } be a sequence in B ( ¯ x ; δ ) ∩ D = ( ¯ x − δ , ¯ x + δ ) ∩ D that converges to ¯ x and x n 6 = ¯ x for all n . By Theorem 3.1.2 , lim n → ∞ f ( x n ) = ` 1 and lim n → ∞ g ( x n ) = ` 2 . Since f ( x n ) ≤ g ( x n ) for all n ∈ N , applying Theorem 2.1.5 , we obtain ` 1 ≤ ` 2 . Theorem 3.1.6 Let f , g : D → R and let ¯ x be a limit point of D . Suppose lim x → ¯ x f ( x ) = ` 1 , lim x → ¯ x g ( x ) = ` 2 , and ` 1 < ` 2 . Then there exists δ > 0 such that f ( x ) < g ( x ) for all x ∈ B ( ¯ x ; δ ) ∩ D , x 6 = ¯ x .