Sequences of Real Numbers, Convergence
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The chapter begins with a discussion of sequences that culminates in the concept of convergence, the fundamental concept of analysis. The Weierstrass-Bolzano theorem (§3.5), nominally a theorem about bounded sequences, is in essence a property of closed intervals; Cauchy’ criterion (§3.6) is a test for convergence, especially useful in the theory of infinite series (§10.1). The chapter concludes with a dissection of convergence into two more general limiting operations.
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