Mathematical Analysis I, 2004a

§§8–9. Upper and Lower Bounds. Completeness Axiom 37
so q = q ′ after all. Uniqueness of min A is proved in the same manner.
Note 3. If A has one lower bound p, ithasmany (e.g., take any p ′ q).
Geometrically, on the real axis, all lower (upper) bounds lie to the left (right)
of A; seeFigure 1.
p ′
p
A
{ }} {
q q ′
u
Figure 1
v
Examples.
(1) Let
A = {1, −2, 7}.
Then A is bounded above (e.g., by 7, 8, 10, ...) and below (e.g., by
−2, −5, −12, ...).
We have min A = −2, max A =7.
(2) The set N of all naturals is bounded below (e.g., by 1, 0, 1 2
, −1, ...),
and 1 = min N; N has no maximum, for each q ∈ N is exceeded by some
n ∈ N (e.g., n = q +1).
(3) Given a, b ∈ F (a ≤ b), we define in F the open interval
the closed interval
the half-open interval
and the half-closed interval
(a, b) ={x | a