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This programme uses the completeness axiom to demonstrate that any arbitrary bounded set of real numbers has a smallest closed interval which contains it.
Metadata describing this Open University video programme
Module code and title: M203, Introduction to pure mathematics
Item code: M203; 08
First transmission date: 10-04-1980
Published: 1980
Rights Statement:
Restrictions on use:
Duration: 00:20:45
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Producer: David Saunders
Contributors: Timothy R. Lister; Colin Rourke
Publisher: BBC Open University
Keyword(s): Bisection; Closed Interval; Least Closed Interval; Nests; Real Numbers
Footage description: Colin Rourke explains that he is going to try and show that an arbitary bounded set of real numbers has a smallest closed interval which contains it. He defines a closed interval and nests of intervals. The bisection process for root 2 produces a sequence of nested intervals. A computer animation is used to explain the bisection process. Colin Rourke then discusses the completeness axiom which states that a sequence of nested intervals like the one shown always contains a unique real number. He now works out the smallest closed interval for a specific set of numbers. Tim Lister then gives a general proof that any bounded set of real numbers has a smallest closed interval. First of all he shows that there is not a smallest closed interval for a set of rational numbers. He then uses the completeness axiom twice to construct a particular closed interval (...). These endpoints are constructed by the use of the bisection process. This process is shown as a computer animation. Having constructed the end points he shows that the points do contain a set and that it is the smallest interval that does. To do this he argues by contradiction. The assumptions made in constructing the end points ensure that these points do fulfill these conditions. Colin Rourke sums up the results of the programme.
Master spool number: HOU3144
Production number: FOUM018S
Videofinder number: 4095
Available to public: no