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n=1 2m-1 m Let {} be the 'interlaced' sequence a, b, a, b defined by c₂ = a, c. = b Show that lim supa is the largest subsequence limit of {a}. Show that (1) there is some subsequence of {a} which converges to lim supa, but (2) if a > lim sup a then there is no subsequence of {a} which converges to a. Assume lim sup a = ± 00 -00 11-0011

n=1 2m-1 m Let {} be the 'interlaced' sequence a, b, a, b defined by c₂ = a, c. = b Show that lim supa is the largest subsequence limit of {a}. Show that (1) there is some subsequence of {a} which converges to lim supa, but (2) if a > lim sup a then there is no subsequence of {a} which converges to a. Assume lim sup a = ± 00 -00 11-0011

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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Let \((c_n)_{n=1}^{\infty}\) be the 'interlaced' sequence \(a_1, b_1, a_2, b_2, \ldots\) defined by \(c_{2m-1} = a_m\), \(c_{2m} = b_m\). Show that \(\limsup_{n \to \infty} a_n\) is the largest subsequence limit of \(\{a_n\}\). Show that (1) there is some subsequence of \(\{a_n\}\) which converges to \(\limsup_{n \to \infty} a_n\), but (2) if \(\alpha > \limsup_{n \to \infty} a_n\) then there is no subsequence of \(\{a_n\}\) which converges to \(\alpha\). Assume \(\limsup_{n \to \infty} a_n \neq \pm \infty\).
Transcribed Image Text:Let \((c_n)_{n=1}^{\infty}\) be the 'interlaced' sequence \(a_1, b_1, a_2, b_2, \ldots\) defined by \(c_{2m-1} = a_m\), \(c_{2m} = b_m\). Show that \(\limsup_{n \to \infty} a_n\) is the largest subsequence limit of \(\{a_n\}\). Show that (1) there is some subsequence of \(\{a_n\}\) which converges to \(\limsup_{n \to \infty} a_n\), but (2) if \(\alpha > \limsup_{n \to \infty} a_n\) then there is no subsequence of \(\{a_n\}\) which converges to \(\alpha\). Assume \(\limsup_{n \to \infty} a_n \neq \pm \infty\).
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