Section23supplementTimeless

Section23supplementTimeless - Econ 204 Supplement to...

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Unformatted text preview: Econ 204 Supplement to Section 2.3 Lim Sup and Lim Inf Deﬁnition 1 We extend the deﬁnition of sup and inf to unbounded sets as follows: sup S = +∞ if S is not bounded above inf S = −∞ if S is not bounded below Deﬁnition 2 [Deﬁnition 3.7 in de La Fuente] If {xn } is a sequence of real numbers, we say that {xn } tends to inﬁnity (written {xn } → ∞ or limn→∞ xn = ∞) if ∀ K ∈ R ∃ N ( K ) n > N (K ) ⇒ x n > K Similarly, we say limn→∞ xn = −∞ if ∀ K ∈ R ∃ N ( K ) n > N (K ) ⇒ x n < K Deﬁnition 3 Consider a sequence {xn } of real numbers. Let αn = sup{xk : k ≥ n} = sup{xn , xn+1 , xn+2 , . . .} βn = inf {xk : k ≥ n} Notice that either αn = ∞ for all n; or αn is a decreasing sequence of real numbers, in which case αn tends to a limit (either a real number or −∞) by Theorem 3.1 and Deﬁnition 3.7 ; similarly, either βn = −∞ for all n; or βn is a increasing sequence of real numbers; in which case βn tends to a limit (either a real number or ∞). Thus, we deﬁne lim sup xn = n→∞ lim inf xn = n→∞ +∞ limn→∞ αn if αn = +∞ for all n otherwise −∞ if βn = −∞ for all n limn→∞ βn otherwise 1 Theorem 4 Let {xn } be a sequence of real numbers. Then lim xn = x ∈ R ∪ {−∞, ∞} n→∞ if and only if lim inf xn = lim sup xn = x n→∞ n→∞ 2 ...
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This note was uploaded on 08/01/2008 for the course ECON 204 taught by Professor Anderson during the Fall '08 term at Berkeley.

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Section23supplementTimeless - Econ 204 Supplement to...

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