Limits Of Sets

Increasing Sequence of Sets

Let

(A_{n})

be a sequence of sets. The sequence

(A_{n})

is increasing if

A_{1} \subseteq A_{2} \subseteq A_{3} \subseteq \dots .

Decreasing Sequence of Sets

Let

(A_{n})

be a sequence of sets. The sequence

(A_{n})

is decreasing if

A_{1} \supseteq A_{2} \supseteq A_{3} \supseteq \dots .

Sequence of Sets Increases to a Set

Let

(A_{n})

be an increasing sequence of sets. We say that

(A_{n})

increases to

A

, and write

A_{n} ↗ A

, if

A = ⋃_{n = 1}^{\infty} A_{n} .

Sequence of Sets Decreases to a Set

Let

(A_{n})

be a decreasing sequence of sets. We say that

(A_{n})

decreases to

A

, and write

A_{n} ↘ A

, if

A = ⋂_{n = 1}^{\infty} A_{n} .

The intervals $A_{n} = [- 1 / n, 1 / n]$ form a decreasing sequence of sets, and $A_{n} ↘ {0}$ . Each later interval is contained in the previous one, and the only point that remains in every interval is $0$ .

Sequence of Sets Shrinks to a Point

Let

X

be a set, let

p \in X

, and let

(A_{n})

be a sequence of subsets of

X

. We say that

(A_{n})

shrinks to

p

A_{n} ↘ {p}

This notation is useful when a quantity is first measured on a region and then localized to one point. For instance, instead of writing an informal limit like $A \to p$ , one can choose sets $A_{n}$ with $A_{n} ↘ {p}$ and then take an ordinary sequence limit as $n \to \infty$ .