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The Nested Interval Theorem

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Archimedes Tube

Published on Nov 10, 2021

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The THEOREM OF BOXED INTERVALS is a result that will allow us to build real numbers from families of intervals. To prove it we will use the SUPREME PRINCIPLE and in fact both are equivalent. In this way, the theorem of boxed intervals could be adopted alternatively as an axiom of real numbers and is commonly known as CANTOR'S AXIOM since it was formulated by the brilliant Russian-German mathematician George Cantor in 1872. An interesting application is the following: The Greeks already knew that there is no rational number whose square is equal to 2. That is, square root of 2 is not a rational number, but is there any real number whose square is 2? Is square root of 2 a real number? We are so used to saying that square root of 2 is irrational and therefore, a real number, that we have not realized that this statement has to be demonstrated. In this video we will demonstrate it thanks to the THEOREM OF BOXED INTERVALS.

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