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The Limit of a Sequence - MIT Mathematics

3 The Limit of a Definition of Chapter 1 we discussed the Limit of sequences that were monotone; thisrestriction allowed some short-cuts and gave a quick introduction to the many important sequences are not monotone numerical methods, for in-stance, often lead to sequences which approach the desired answer alternatelyfrom above and below. For such sequences, the methods we used in Chapter 1won t work. For instance, the , .9, , .99, , .999, ..has 1 as its Limit , yet neither the integer part nor any of the decimal places of thenumbers in the Sequence eventually becomes constant. We need a more generallyapplicable definition of the abandon therefore the decimal expansions, and replace them by the ap-proximation viewpoint, in which the Limit of{an}isL means roughlyanis a good approximation toL, whennis following definition makes this precise. After the definition, most of therest of the chapter will consist of examples in which the Limit of a Sequence iscalculated directly from this definition.

“obvious” using the definition of limit we started with in Chapter 1, but we are committed now and for the rest of the book to using the newer Definition 3.1 of limit, and therefore the theorem requires proof. Theorem 3.2B {an} increasing, L = liman ⇒ an ≤ L for all n; {an} decreasing, L = liman ⇒ an ≥ L for all n. Proof.

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