Nested Intervals
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals I_n on the real number line with natural numbers n=1,2,3,\dots as an index. In order for a sequence of intervals to be considered nested intervals, two conditions have to be met: # Every interval in the sequence is contained in the previous one (I_ is always a subset of I_n). # The length of the intervals get arbitrarily small (meaning the length falls below every possible threshold \varepsilon after a certain index N). In other words, the left bound of the interval I_n can only increase (a_\geq a_n), and the right bound can only decrease (b_\leq b_n). Historically - long before anyone defined nested intervals in a textbook - people implicitly constructed such nestings for concrete calculation purposes. For example, the ancient Babylonians discovered a method for computing square roots of numbers. In contrast, the famed Archimedes constructed sequences of polygon ...
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Illustration Nested Intervals
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