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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.
 Fall '08
 ANDERSON

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