Editing Set-theoretic limit (section)

===Monotone sequences===
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The sequence <math>\left(A_n\right)</math> is said to be '''nonincreasing''' if <math>A_{n+1} \subseteq A_n</math> for each <math>n,</math> and '''nondecreasing''' if <math>A_n \subseteq A_{n+1}</math> for each <math>n.</math> In each of these cases the set limit exists. Consider, for example, a nonincreasing sequence <math>\left(A_n\right).</math> Then 
<math display=block>\bigcap_{j \geq n} A_j = \bigcap_{j \geq 1} A_j \text{ and } \bigcup_{j \geq n} A_j = A_n.</math>
From these it follows that
<math display=block>\liminf_{n \to \infty} A_n = \bigcup_{n \geq 1} \bigcap_{j \geq n} A_j = \bigcap_{j \geq 1} A_j = \bigcap_{n \geq 1} \bigcup_{j \geq n} A_j = \limsup_{n \to \infty} A_n.</math>
Similarly, if <math>\left(A_n\right)</math> is nondecreasing then 
<math display=block>\lim_{n \to \infty} A_n = \bigcup_{j \geq 1} A_j.</math>

The [[Cantor set#Construction and formula of the ternary set|Cantor set]] is defined this way.