Page 48 - 'Blast_Into_Math

Blast into Math! ets of nummers: mathematical plaagrounds

By definition of upper bound, all elements of S are less than or equal to q , and since |x|≥ q , |x| is
also an upper bound for S .

Now, let’s look for the largest element of S. Since S = ∅ , there is some

s 1 ∈ S.

Then, either s 1 is the largest element of S or not. If not, there is some

s 2 ∈ S, with s 2 >s 1 .

Then, either s 2 is the largest element of S or not. We can continue looking for bigger and bigger

elements of S . But, when will we stop? Is it possible that we could keep looking for bigger and bigger
elements of S forever? Let’s think about this. Since s 1 ∈ S , and the integers are closed under addition
and subtraction,

|x|− s 1 ∈ Z, and |x|− s 1 ≥ 0because s 1 ≤|x|.

Since s 2 ∈ Z , and s 2 >s 1 this means that s 2 ≥ s 1 +1. Because |x| is an upper bound for S ,
s 2 ≤|x| . So, we can do the same thing again, finding s 3 and s 4 ,

s 1 <s 1 +1 ≤ s 2 <s 2 +1 ≤ s 3 < ... ≤|x|.

Since each new larger element of S is at least 1 larger than the previous, there are at most |x|− s 1
integers between s 1 and |x|. This means that there are at most this many elements of S between s 1
and |x| , because S contains only integers. So listing the elements of S starting from s 1 and increasing,
we will reach the largest element of S .

The unique largest element of the set S is its least upper bound.

Definition 3.3.3 Let S be a non-empty set of rational numbers which is bounded above. If there exists
an upper bound A which is less than or equal to all other upper bounds of S , then A is called the least
upper bound (or LUB) of S.

The least upper bound is exactly that: it is the smallest or least upper bound.

Proposition 3.3.4 (GLB Proposition (LUB-Twin Proposition)) Any non-empty set of integers which is
bounded below contains a unique smallest element.

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