Page 47 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

Definition 3.3.1 Let S be a non-empty set of rational numbers. If there is some x ∈ Q such that every

element of S is less than or equal to x , then S is bounded above, and x is called an upper bound for S.
If there is some y ∈ Q such that every element of S is greater than or equal to y , then S is bounded
below, and y is called a lower bound for S . If S is both bounded above and bounded below, then S is
bounded.

The concept of bounded is similar to the rules of certain games like tennis and soccer, in which the ball
must stay in bounds. The elements of a bounded set are like balls which must stay within the bounds.
If we use the number line, then when a set is bounded above, there is some x on the number line such
that the entire set is to the left of x. So, in this game the ball cannot go past x . If the set is bounded
below, this means there is some y on the number line such that the entire set is to the right of y . So,
in this game, the ball cannot go past y . When a set is bounded, the whole set is between x and y; the
ball must stay between x and y . GO TEAM!

Exercise: In the picture, what are the upper and lower bounds?

Proposition 3.3.2 (LUB Proposition) Any non-empty set of integers which is bounded above contains a
unique largest element.

Proof: Let S be a non-empty set of integers which is bounded above by some q ∈ Q . This means that
every element of S is less than or equal to q . Since S is a set of integers, it will be easier to compare
the elements of S to an integer. The rational number q ∈ Q , so by the Rational Theorem there is
x ∈ Z and y ∈ N such that

x
q = .
y

Since x ≤|x| and y ∈ N means that y ≥ 1,

|x|
q ≤ ≤|x|.
y

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