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growing_sequence [2013/08/09 16:02] nikolaj |
growing_sequence [2014/03/21 11:11] |
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- | ===== Growing sequence ===== | ||
- | ==== Definition ==== | ||
- | | @#88DDEE: $X$ | | ||
- | | @#55EE55: $A\in \mathrm{GrowingSequence}(X) $ | | ||
- | |||
- | | @#88DDEE: $A\in \mathrm{Sequence}(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 === | ||
- | | @#FFBB00: $A_n\uparrow \hat A \equiv (A\in \mathrm{GrowingSequence}(X))\land(\lim_{n\to\infty}A_n=\hat A)$ | | ||
- | ==== Context ==== | ||
- | === Subset of === | ||
- | [[Sequence]] |