8

Question 1

The triangle inequality

Answer 1

Ix + yI ≤ IxI + IyI

Consequences:

• Ix - yI ≤ Ix - zI + Iy - zI
  (because Ix - yI = I(x - z) + (z - y)I ≤ Ix - zI + Iz - yI)
• for any two real numbers x, y,
  Ix - yI ≥ IIxI - IyII

Question 2

S ⊆ R is bounded above if and only if

Nb A⊆B means A has some or all elements of B

Answer 2

there is M ∈ R such that
for all x ∈ S; x ≤ M.
In this case, M is called an upper bound of/for/on S.

Question 3

S ⊆ R is bounded below if and only if

Answer 3

there is m ∈ R such that
for all x ∈ S, x ≥ m.
In this case, m is called a lower bound of/on/for S.

Question 4

A set is said simply to be bounded if it is

Answer 4

bounded above and bounded below

Question 5

The Continuum property:

Answer 5

Every non-empty set of real numbers that is bounded above has a least upper bound

and any non-empty set of real numbers that is bounded below has a greatest lower bound.

For a non-empty subset S of R, if S is bounded above, the least upper bound of S (which exists, by the continuum property) is called the supremum of S, and
is denoted sup S. If S is bounded below, the greatest lower bound is called the infimum of S and is denoted inf S.
Note that the supremum and infimum of a set S (when they exist; i.e. when S is bounded above/below) need not belong to S.