Mathematical Bridges by Titu Andreescu Cristinel Mortici & Marian Tetiva

Author:Titu Andreescu, Cristinel Mortici & Marian Tetiva
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(where the infimum can be −∞). The monotone convergence theorem gives further information, namely, that every bounded monotone sequence of real numbers is convergent, i.e., it surely has a finite limit. The proof is based on the least upper bound axiom, asserting that any nonempty bounded above set of real numbers has a least upper bound. (Actually this statement is equivalent to the nested intervals theorem that follows, and each of them expresses the completeness of the system of real numbers.)

As a direct consequence of the monotone convergence theorem, we give the following result. We note that if I = [a, b] is a closed interval, then l(I) = b − a denotes the length of I.

The Nested Intervals Theorem.

Let

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