Monotonous sequence of quantities

A monotonous sequence of sets is a special sequence of sets in which special inclusion relationships apply. If a set with a smaller index is always contained in a set with a larger index, then the sequence is called a monotonically increasing set sequence . If a set with a smaller index always contains a set with a larger index, the sequence is called a monotonically decreasing sequence of sets . Monotonous set sequences can be understood as a special case of a monotonous mapping .

definition

A sequence of sets is called ${\ displaystyle (A_ {i}) _ {i \ in \ mathbb {N}}}$

Monotonically increasing or monotonically increasing , if applies. ${\ displaystyle A_ {k} \ subseteq A_ {k + 1}}$
Monotonously decreasing if applies. ${\ displaystyle A_ {k} \ supseteq A_ {k + 1}}$
Monotone if it is either monotonically increasing or monotonically decreasing.

Sometimes there is also the designation of a monotonically increasing set sequence or a monotonically decreasing set sequence .

Examples

The sequence of quantities defined by

{\ displaystyle (A_ {k}) _ {k \ in \ mathbb {N}} = \ {0, \ dots, k \} \ subset \ mathbb {N}}

is a monotonically increasing set sequence, since every set contains all elements of the set .

{\ displaystyle A_ {k}}

{\ displaystyle A_ {k-1}}

The sequence is monotonically increasing. This follows directly from the monotony of the real sequence . ${\ displaystyle (A_ {k}) _ {k \ in \ mathbb {N}} = [0.1-1 / k] _ {k \ in \ mathbb {N}} = (\ {0 \}, [ 0.1 / 2], [0.2 / 3], \ dotsc)}$ ${\ displaystyle (a_ {k}) _ {k \ in \ mathbb {N}} = \ left ({\ tfrac {1} {k}} \ right) _ {k \ in \ mathbb {N}}}$
Likewise, the sequence is monotonically decreasing. ${\ displaystyle (A_ {k}) _ {k \ in \ mathbb {N}} = [0.1 / k] _ {k \ in \ mathbb {N}} = ([0.1], [0, 1/2], [0,1 / 3], \ dotsc)}$