Mathematical Analysis I, 2004a

§8. Sequences 17
Definition 1.
A real sequence {u n } is said to be monotone (or monotonic) iffitiseither
nondecreasing, i.e.,
(∀ n) u n ≤ u n+1 ,
or nonincreasing, i.e.,
(∀ n) u n ≥ u n+1 .
Notation: {u n }↑ and {u n }↓, respectively. If instead we have the strict
inequalities u n u n+1 ), we call {u n } strictly
monotone (increasing or decreasing).
A similar definition applies to sequences of sets.
Definition 2.
A sequence of sets A 1 ,A 2 , ..., A n , ... is said to be monotone iff it is
either expanding, i.e.,
or contracting, i.e.,
(∀ n) A n ⊆ A n+1 ,
(∀ n) A n ⊇ A n+1 .
Notation: {A n }↑ and {A n }↓, respectively. For example, any sequence of
concentric solid spheres (treated as sets of points), with increasing radii,
is expanding; if the radii decrease, we obtain a contracting sequence.
Definition 3.
Let {u n } be any sequence, and let
n 1