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