Guide for Math 311

1.1.1. Definitions 1. Bounded Above (for a set): Let S ⊂ R. S is bounded above if ∃L ∈ R such that x ≤ L for all x ∈ S. L is called an upper bound for S. (Similar definition for bounded below and lower bound) 2. Bounded (for a set): Let S be a set. S is bounded if S is bounded above and bounded below. 3. Least Upper Bound (LUB) Axiom for R (for a set): Let S ⊂ R be bounded above. Then !∃L ∈ R such that (a) x ≤ L for all x ∈ S (i.e., L is an upper bound for S) (b) if L′ < L, then ∃x ∈ S such that x > L′. Alternatively, given ǫ > 0, ∃xǫ ∈ S such that L− ǫ < xǫ. (i.e., L is the least upper bound for S) L is called the least upper bound of S and is denoted by supS. (Similar statement for Greatest Lower Bound (GLB) Axiom for R and inf S) 4. Sequence: Let S ⊂ R be nonempty. A sequence s in S is (a) an ordered list with entries in S, i.e., s = (s1, s2, ..., sk, ...), where sk is the entry in the kth position and every s ∈ S. We write s = {sk} ∞ k=1 or s = {sk}. (b) a function from N into S.