1. Index FAQ Limits of Sequences of Real Numbers Sequences of Real Numbers Limits through Rigorous Definitions The Squeeze Theorem Using the Squeeze Theorem Monotonous Sequences

2. Index FAQ Sequences of Numbers Definition Examples 1 2 3

3. Index FAQ Limits of Sequences Definition Examples1 If a sequence has a finite limit, then we say that the sequence is convergent or that it converges. Otherwise it diverges and is divergent. 0

4. Index FAQ 0

5. Index FAQ Limits of Sequences 2 3 Notation The sequence (1,-2,3,-4,…) diverges.

6. Index FAQ Computing Limits of Sequences (1)

7. Index FAQ Computing Limits of Sequences (1) Examples 1 2 1 n2n2 0

8. Index FAQ Computing Limits of Sequences Examples continued 3

9. Index FAQ Formal Definition of Limits of Sequences Definition Example

10. Index FAQ Limit of Sums Theorem Proof

11. Index FAQ Limit of Sums Proof By the Triangle Inequality

12. Index FAQ Limits of Products The same argument as for sums can be used to prove the following result. Theorem Remark Examples

13. Index FAQ Squeeze Theorem for Sequences Theorem Proof

14. Index FAQ Using the Squeeze Theorem Example Solution This is difficult to compute using the standard methods because n! is defined only if n is a natural number. So the values of the sequence in question are not given by an elementary function to which we could apply tricks like L’Hospital’s Rule. Here each term k/n < 1.

15. Index FAQ Using the Squeeze Theorem Problem Solution

16. Index FAQ Monotonous Sequences Definition The sequence (a 1,a 2,a 3,…) is decreasing if a n+1 ≤ a n for all n. A sequence (a 1,a 2,a 3,…) is increasing if a n ≤ a n+1 for all n. The sequence (a 1,a 2,a 3,…) is monotonous if it is either increasing or decreasing. Theorem The sequence (a 1,a 2,a 3,…) is bounded if there are numbers M and m such that m ≤ a n ≤ M for all n. A bounded monotonous sequence always has a finite limit. Observe that it suffices to show that the theorem for increasing sequences (a n ) since if (a n ) is decreasing, then consider the increasing sequence (-a n ).

17. Index FAQ Monotonous Sequences Theorem A bounded monotonous sequence always has a finite limit. Proof Let (a 1,a 2,a 3,…) be an increasing bounded sequence. Then the set {a 1,a 2,a 3,…} is bounded from the above. By the fact that the set of real numbers is complete, s=sup {a 1,a 2,a 3,…} is finite. Claim

18. Index FAQ Monotonous Sequences Theorem A bounded monotonous sequence always has a finite limit. Proof Let (a 1,a 2,a 3,…) be an increasing bounded sequence. Let s=sup {a 1,a 2,a 3,…}. Claim Proof of the Claim