1. Introduction to Real Analysis Dr. Weihu Hong Clayton State University 9/18/2008

2. Theorem 2.3.2 If is monotone and bounded, then it converges. Corollary 2.3.3 If is a sequence of closed and bounded intervals with for all n єN, then Note: The intervals must be closed in Corollary 2.3.3

3. Infinite Limits Definition 2.3.6 Let be a sequence of real numbers. We say that approaches infinity, or that diverges to ∞, denoted if for every positive real number M, there exists an integer KєN such that How would you define a sequence approaches to −∞?

4. Theorem 2.3.7 If is monotone increasing and not bounded above, then Proof: Since the sequence is not bounded above, therefore, for every positive number M, there exists a term such that Since the sequence is increasing, thus, Therefore,

5. Subsequence of a sequence Definition 2.4.1 Given a sequence in R, consider a sequence of positive integers such that. Then the sequence is called a subsequence of the sequence.

6. Examples of subsequences of a sequence Consider a sequence. Let be two sequences of positive integers. Then we have two subsequences and of the sequence, one of which is consisting of all the terms from the sequence with odd indices while the other one is consisting of all the terms from the sequence with even indices.

7. Subsequential limit of a sequence Given a sequence.Let a be either a real number or ±∞. We say that a is a subsequential limit of the sequence if there exists a subsequence such that

8. Example of subsequential limit Consider the sequence. Is a = 2 a subsequential limit of the sequence? Is a = 0 a subsequential limit of the sequence? Consider the sequence. Is a = +∞ a subsequential limit of the sequence? Is a = -∞ a subsequential limit of the sequence?

9. Theorem 2.4.3 Let be a sequence in R. If converges to p, then every subsequence of also converges to p.