In a typical calculus course, sequences are usually treated very briefly, and their role is primarily as a prelude to the study of series. In real analysis, by contrast, sequences assume a much more important role. We will certainly use sequences in our study of series in Chapter 9, but, as will be seen in the present chapter, we will prove some substantial and important theorems about sequences in their own right, such as the Monotone Convergence Theorem (Corollary 8.3.4) and the Bolzano–Weierstrass Theorem (Theorem 8.3.9). As was the case for the important theorems concerning continuity, derivatives and integrals that we saw in previous chapters, the important theorems concerning sequences rely upon the Least Upper Bound Property of the real numbers.
KeywordsCauchy Sequence Convergent Subsequence Real Analysis Great Element Fibonacci Number
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