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growing_sequence [2013/09/08 15:43] nikolaj |
growing_sequence [2014/03/21 11:11] |
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| - | ===== Growing sequence ===== | ||
| - | ==== Definition ==== | ||
| - | | @#88DDEE: $X$ | | ||
| - | | @#FFBB00: $A\in \mathrm{GrowingSequence}(X) $ | | ||
| - | |||
| - | | @#55EE55: $A\in \mathrm{InfSequence}(X) $ | | ||
| - | |||
| - | | $n\in \mathbb N$ | | ||
| - | |||
| - | | @#55EE55: $A_{n}\subseteq A_{n+1} $ | | ||
| - | |||
| - | ==== Discussion ==== | ||
| - | |||
| - | === Ramifications === | ||
| - | For [[Falling sequence|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 === | ||
| - | | @#EEEE55: $A_n\uparrow \hat A \equiv A\in \mathrm{GrowingSequence}(X)\ \land\ \lim_{n\to\infty}A_n=\hat A$ | | ||
| - | ==== Parents ==== | ||
| - | === Subset of === | ||
| - | [[Infinite sequence]] | ||
