1. Introduction to Real Analysis Dr. Weihu Hong Clayton State University 9/3/2009

2. Properties on the absolute value |x| = x if x>0; -x if x≤0. The geometric meaning of |x|: the distance between x and 0. Theorem 2.1.2 (Properties on |x|) |-x| = |x| for all x єR |xy| =|x||y| for all x, y єR |x| = for all xєR If r > 0, then |x| < r if and only if –r < x < r. -|x| ≤ x ≤ |x| for all xєR You should know how to prove each of these statements!

3. Properties on the absolute value Theorem 2.1.3 (Triangle inequality) For all x, y єR, we have |x + y| ≤ |x| + |y|. Proof: Consider 0 ≤ (x + y)² Corollary 2.1.4 For all x, y, zєR, we have |x – y|≤ |x – z| + |z – y| ||x| - |y|| ≤ |x – y| Make sure you know how to prove these statements!

4. Neighborhood of a point Definition 2.1.6 (ε-neighborhood of the point p) Let pєR and let ε > 0. The set is called an ε-neighborhood of the point p. Note: it is the same if we write

5. Convergence of a Sequence Definition 2.1.7 A sequence in R is said to converge if there exists p єR such that for every ε>0, there exists a positive integer K such that for all n≥K. In this case, we say that converges to p, or that p is the limit of the sequence, and we write If does not converge, then is said to diverge.

6. Bounded Sequence A sequence in R is said to be bounded if there exists a positive constant M such that for all n єN. How would you define a sequence is unbounded? Theorem 2.1.10 (a) If a sequence in R converges, then its limit is unique. (b) Every convergent sequence in R is bounded. You need to know how to prove both of these!

7. More theorems on limit Theorem 2.2.1 If are convergent sequences of real numbers with then

8. More theorems on limit Corollary 2.2.2 If is a convergent sequence of real numbers with then for any c єR,

9. More theorems on limit Theorem 2.2.3 Let be sequences of real numbers with then

10. More theorems on limit Theorem 2.2.4 (Squeeze theorem) Let be sequences of real numbers for which there exists K єN such that then