2. Theorem 1-a Let be a sequence: L L a) converges to a real number L iff every subsequence of converge to L Illustrations 0 0

3. Theorem 1-b L. If L is a limit of, then L is the only limit of. Illustrations

4. Example 1

5. Example 2

6. Example 3

7. Theorem 1-c c) If is eventually in Illustration

8. d) If is eventually in Illustration Theorem 1-d

9. Theorem 2 Let be a sequence: a)If is bounded from above and increasing then it converge to the supremum of the range of. Illustration 5 It is bounded from above & increasing It converges to the sup of its range, which is 5

10. Question

12. Theorem 2 b) If is bounded from below and decreasing then it converges to the infimum of the range of. Illustration 0 It is bounded from below & decreasing It converges to the inf of its range which is 0

13. Theorem 3 A convergent sequence is bounded

14. Operations On Convergent Sequences

15. Illustrations Find

16. Examples

17. Solutions

18. Solutionss

21. Question