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Prove the lim(Sn) = -∞ case.

Prove the lim(Sn) = -∞ case.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Prove the lim(Sn) = -∞ case.

10.7 Theorem. Let (sn) be a sequence in R. (i) If lim s, is defined [as a real number, +o or -0o), then lim inf s, = lim s, = lim sup s,. (ii) If liminf s, = limsup sn, then lim s, is defined and lim s, = lim inf s, = limsup s,. Proof We use the notation un = inf{sn : n > N}, vN = sup{sn : n > N}, u = lim un = liminf s, and v = lim vN = lim sup s,. (i) Suppose lim sn = +oo. Let M be a positive real number. Then there is a positive integer N so that n > N implies Sn > M. Then uy = inf{s, :n > N} 2 M. It follows that m > N implies um 2 M. In other words, the sequence (uN) satisfies the condition defining lim uy = +o, i.e., lim inf sn = +o. Likewise limsup sn = +0. The case lim s, = -00 is handled in a similar manner.
Transcribed Image Text:10.7 Theorem. Let (sn) be a sequence in R. (i) If lim s, is defined [as a real number, +o or -0o), then lim inf s, = lim s, = lim sup s,. (ii) If liminf s, = limsup sn, then lim s, is defined and lim s, = lim inf s, = limsup s,. Proof We use the notation un = inf{sn : n > N}, vN = sup{sn : n > N}, u = lim un = liminf s, and v = lim vN = lim sup s,. (i) Suppose lim sn = +oo. Let M be a positive real number. Then there is a positive integer N so that n > N implies Sn > M. Then uy = inf{s, :n > N} 2 M. It follows that m > N implies um 2 M. In other words, the sequence (uN) satisfies the condition defining lim uy = +o, i.e., lim inf sn = +o. Likewise limsup sn = +0. The case lim s, = -00 is handled in a similar manner.
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