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(a) Suppose (k)=1 is a sequence in R" and 1 € R". Prove that the following are equivalent: ● (k)1 does not converge to l. • There exists a subsequence (m)_1 of (x)=1 and > 0 so that ||m² − 1|| ≥ & for all k € N.

(a) Suppose (k)=1 is a sequence in R" and 1 € R". Prove that the following are equivalent: ● (k)1 does not converge to l. • There exists a subsequence (m)_1 of (x)=1 and > 0 so that ||m² − 1|| ≥ & for all k € N.

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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3. (a) Suppose (x)_₁ is a sequence in R" and 1 € R". Prove that the following are equivalent: ● (xk)1 does not converge to l. • There exists a subsequence (mk)k=1 of (xk)=1 and € > 0 so that ||Tmk − 1|| ≥ e for all k € N. (b) Suppose (*)_₁ is a bounded sequence in R" and 1 € R". Prove that the following are equivalent: ● (k) converges to l. Every subsequence (™mk)k=1 of (™k)%=1 has a further subsequence (pmp)=1 (sorry for all the subscripts...) such that (pm)-1 converges to l. ●
Transcribed Image Text:3. (a) Suppose (x)_₁ is a sequence in R" and 1 € R". Prove that the following are equivalent: ● (xk)1 does not converge to l. • There exists a subsequence (mk)k=1 of (xk)=1 and € > 0 so that ||Tmk − 1|| ≥ e for all k € N. (b) Suppose (*)_₁ is a bounded sequence in R" and 1 € R". Prove that the following are equivalent: ● (k) converges to l. Every subsequence (™mk)k=1 of (™k)%=1 has a further subsequence (pmp)=1 (sorry for all the subscripts...) such that (pm)-1 converges to l. ●
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