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## Falling sequence

### Definition

 $X$
 $A\in \mathrm{FallingSequence}(X)$
 $A\in \mathrm{InfSequence}(X)$
 $n\in \mathbb N$
 $A_{n+1}\subseteq A_n$

### Discussion

#### Ramifications

For falling sequences we have: $\lim_{n\to\infty}A_n=\bigcap_{n=1}^\infty A_n$.

For growing sequences we have: $\lim_{n\to\infty}A_n=\bigcup_{n=1}^\infty A_n$.

#### Predicates

 $A_n\downarrow \hat A \equiv (A\in \mathrm{FallingSequence}(X))\land(\lim_{n\to\infty}A_n=\hat A)$