Theorem 47. Let (a;) be a sequence with values in an ordered field F, and let c e F. If (a;) converges then (ca;) converges and lim ca; = c lim a¿. Corollary 48. Let (a;) be a sequence with values in an ordered field F. If (a;) converges, then so does (-a;), and lim -a, = - lim a
Theorem 47. Let (a;) be a sequence with values in an ordered field F, and let c e F. If (a;) converges then (ca;) converges and lim ca; = c lim a¿. Corollary 48. Let (a;) be a sequence with values in an ordered field F. If (a;) converges, then so does (-a;), and lim -a, = - lim a
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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Question
Prove Theorem 47.
Prove Corollary 48.
Please answer them separately.
Transcribed Image Text:Exercise 25. Prove the above theorem and corollary. Hint: in the case
that c +0 choose e' = €/|cl. What if c = 0?
Transcribed Image Text:Theorem 47. Let (a;) be a sequence with values in an ordered field F, and
let ce F. If (ai) converges then (ca;) converges and
lim ca; = c lim a;.
i+00
Corollary 48. Let (a;) be a sequence with values in an ordered field F.
If (a;) converges, then so does (-a;), and
lim -a; = - lim a;
i+00
i+00
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