By Jiri Lebl

A primary direction in mathematical research. Covers the true quantity process, sequences and sequence, non-stop capabilities, the by-product, the Riemann vital, sequences of capabilities, and metric areas. initially built to educate Math 444 at college of Illinois at Urbana-Champaign and later superior for Math 521 at collage of Wisconsin-Madison. See http://www.jirka.org/ra/

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Let C := {ab : a ∈ A, b ∈ B}. Show that C is a bounded set and that sup C = (sup A)(sup B) and inf C = (inf A)(inf B). 11 (Hard): Given x > 0 and n ∈ N, show that there exists a unique positive real number r such that x = rn . Usually r is denoted by x1/n . 3. 5-1 lecture A concept we will encounter over and over is the concept of absolute value. You want to think of the absolute value as the “size” of a real number. Let us give a formal definition. |x| := x −x if x ≥ 0, if x < 0. Let us give the main features of the absolute value as a proposition.

We could write this set as {(−1)n : n ∈ N}. When ambiguity could arise, we use the words sequence or set to distinguish the two concepts. Another example of a sequence is the constant sequence. That is a sequence {c} = c, c, c, c, . . consisting of a single constant c ∈ R. ∗ [BS] use the notation (xn ) to denote a sequence instead of {xn }, which is what [R2] uses. Both are common. 39 40 CHAPTER 2. SEQUENCES AND SERIES We now get to the idea of a limit of a sequence. 6 that the notation below is well defined.

If f : D → R and g : D → R are bounded functions and f (x) ≤ g(x) for all x ∈ D, then sup f (x) ≤ sup g(x) x∈D and x∈D inf f (x) ≤ inf g(x). 1) You should be careful with the variables. 1) is different from the x on the right. You should really think of the first inequality as sup f (x) ≤ sup g(y). x∈D y∈D Let us prove this inequality. If b is an upper bound for g(D), then f (x) ≤ g(x) ≤ b and hence b is an upper bound for f (D). Therefore taking the least upper bound we get that for all x f (x) ≤ sup g(y).