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'nested interval theorem'
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| Title of object: |
nested interval theorem |
| Canonical Name: |
NestedIntervalTheorem |
| Type: |
Definition |
| Created on: |
2007-08-06 08:11:45 |
| Modified on: |
2007-08-06 08:11:45 |
| Classification: |
msc:26-00, msc:54C30 |
Preamble:
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Content:
If
$$[a_1,\,b_1]\;\supseteq [a_2,\,b_2]\;\supseteq [a_3,\,b_3]\;\supseteq\ldots$$
is a sequence of nested closed intervals, then
\begin{align}
\bigcap_{n=1}^\infty [a_n,\,b_n] \neq \varnothing.
\end{align}
If also\, $\displaystyle\lim_{n\to\infty}(b_n-a_n) = 0$,\, then the union (1) consists of a unique real number. |
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