Page 159 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

Definition 7.1.4 A sequence of numbers

∞
{x n } n=1

is increasing if,

forall n ∈ N.
x n ≤ x n+1

It is decreasing if,

forall n ∈ N.
x n ≥ x n+1

Both increasing sequences and decreasing sequences are called monotone.

An increasing sequence is like a trail of mathematical ants on the number line marching to the right,
whereas a decreasing sequence is like a trail of mathematical ants on the number line marching to the
left. Where are they going? Since sequences are ordered sets, we can use the concepts from set theory
to understand sequences.

Definition 7.1.5 A sequence

∞
{x n } n=1

is bounded above if there exists P ∈ Q such that

x n ≤ P, ∀n ∈ N.

P is called an upper bound. If there exists an upper bound X such that for any upper bound Y ,

X ≤ Y,

then X is the least upper bound. If there exists Q ∈ Q such that

x n ≥ Q, ∀n ∈ N,

then the sequence is bounded below, and Q is a lower bound. If there exists a lower bound Y such that
for any lower bound Z

Y ≥ Z,

then Y is the greatest lower bound.

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