Page 50 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

Definition 3.3.6 A non-empty set S has the least upper bound property if every non-empty subset of S
which is bounded above has a least upper bound in S . S has the greatest lower bound property if every
non-empty subset which is bounded below has a greatest lower bound in S .

I affectionately call the following theorem lubglub, which reminds me of drinking lemonade on a hot
summer day at a baseball game. What does lubglb sound like to you?

Theorem 3.3.7 (LUBGLB). The integers Z have both the LUB and GLB properties.

Proof: By the LUB Proposition, every non-empty subset of Z which is bounded above contains a unique

largest element, which we’ll call l. This element is the least upper bound because if x is another upper
bound, then it must be larger than every element of the set and so

l ≤ x, forany upper bound x.

Since l is an element of a subset of Z , l ∈ Z .

By the GLB Proposition, every non-empty subset of Z which is bounded below contains a unique
smallest element, which we’ll call g . This element is the greatest lower bound because if y is another
lower bound, then it must be smaller than every element of the set and so

y ≤ g, forany lowerbound y.

Since g is an element of a subset of Z , g ∈ Z .

Unlike the integers, the rational numbers have neither the LUB nor GLB properties, which we’ll prove
later in this book. By the end of this book you will learn about the real numbers, a much larger set of
numbers which contains Q and has the LUB and GLB properties.

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