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falling_sequence [2013/09/08 15:41] nikolaj |
falling_sequence [2014/03/21 11:11] |
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- | ===== Falling sequence ===== | ||
- | ==== Definition ==== | ||
- | | @#88DDEE: $X$ | | ||
- | |||
- | | @#FFBB00: $A\in \mathrm{FallingSequence}(X) $ | | ||
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- | | @#55EE55: $A\in \mathrm{InfSequence}(X) $ | | ||
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- | | $n\in \mathbb N$ | | ||
- | |||
- | | @#55EE55: $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 sequence|growing sequences]] we have: $\lim_{n\to\infty}A_n=\bigcup_{n=1}^\infty A_n$. | ||
- | === Predicates === | ||
- | | @#FFBB00: $A_n\downarrow \hat A \equiv (A\in \mathrm{FallingSequence}(X))\land(\lim_{n\to\infty}A_n=\hat A)$ | | ||
- | ==== Parents ==== | ||
- | === Subset of === | ||
- | [[Infinite sequence]] | ||