![]() 5) Prove that every sequence of real numbers contains a monotone subsequence. Because they lie on a number line, their size can be compared. First of all, if we have defined real numbers as infinite decimals, then the procedure just outlined really does unambiguously define a real number. The sequences we saw in the last section we were usu- 1. Mathematicians also play with some special numbers that aren't Real Numbers. Other important examples of sequences include ones made up of rational numbers, real numbers, and complex numbers. Answer (1 of 9): I can think only of one right now, but maybe I can think of some more in another time. (b) Every bounded sequence of real numbers has at least one subsequen-tial limit. The distance between two real numbers is the absolute Example Define a sequence An example should make this clear. Examples: General term of arithmetic progression: The general term of an arithmetic progression with first term and common difference is: Example 3: Find the general term for the arithmetic sequence and then find. For given" > 0, we wish to find a positive integer N such that n > N implies |1 n − 0| 0 there exists N ∈ N such that Explain the sequences of real numbers In Chapter 2, we developed the equation \(1 + x + x^2 + x^3 + \cdots = \frac that converges to L. Usually (but not always) the sequences that arise in practice have a recog-nisable pattern and can be described by a formula. Let (a n) and (b n) be monotone sequences. ![]()
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