Engineering Calculus Notes 330

Engineering Calculus Notes 330 - 11 Proof. If S is bounded,...

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318 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION 3. any set consisting of a convergent sequence s i together with its limit, or any set consisting of a sequence together with all of its accumulation points. We also want to formulate the idea of a bounded set in R 3 . We cannot talk about such a set being “bounded above” or “bounded below”; the appropriate deFnition is Defnition 3.6.4. A set S R 3 is bounded if the set of lengths of elements of S {b s b | s S } is bounded—that is, if there exists M R such that b s b ≤ M for all s S. (This is the same as saying that there exists some ball B ε ( O ) —where ε > 0 is in general not assumed small—which contains S .) A basic and important property of R 3 is stated in the following. Proposition 3.6.5. For a subset S R 3 , the following are equivalent: 1. S is closed and bounded; 2. S is sequentially compact : every sequence s i of points in S has a subsequence which converges to a point of S . We shall abuse terminology and refer to such sets as compact sets.
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Unformatted text preview: 11 Proof. If S is bounded, then by the Bolzano-Weierstrass Theorem (Proposition 2.3.7 ) every sequence in S has a convergent subsequence, and if S is also closed, then the limit of this subsequence must also be a point of S . Conversely, if S is not bounded , it cannot be sequentially compact since there must exist a sequence s k of points in S with b s k b > k ; such a sequence has no convergent subsequence. Similarly, if S is not closed , there must exist a convergent sequence s k of points in S whose limit L lies outside S ; since every subsequence also converges to L , S cannot be sequentially compact. With these deFnitions, we can formulate and prove the following. 11 The property of being compact has a speciFc deFnition in very general settings; how-ever, in the context of R 3 , this is equivalent to either sequential compactness or being closed and bounded....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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