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February 12th, 2017, 05:10 AM   #1
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Limsup question

Give an example of two bounded sequences

such that limsup n→∞(an+bn)
is not equal to
limsup n→∞ an + limsup n→∞ bn.

Hi guys does this have something to do with the triangle inequality, im struggling to find a sequence which this would work and any hints would be great
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February 12th, 2017, 07:05 AM   #2
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let

$a_n = \sin\left(\dfrac{2 \pi n}{N}\right)$

$b_n = \cos\left(\dfrac{2 \pi n}{N}\right)$

$\limsup\{a_n\} = 1$

$\limsup\{b_n\} = 1$

$\limsup\{a_n + b_n\} = \sqrt{2} \neq 2 = 1+1$
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February 12th, 2017, 07:53 AM   #3
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Hi what is the difference between the "n" and the capital "N"
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February 12th, 2017, 07:58 AM   #4
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I understand that An=cos(n) and Bn=sin(n)
would also work
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February 12th, 2017, 08:00 AM   #5
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Quote:
Originally Posted by Jaket1 View Post
Hi what is the difference between the "n" and the capital "N"
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$n$ is the index, $N$ is just some constant so you can sample the trig functions as finely as you like.
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February 13th, 2017, 05:28 AM   #6
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A simpler example:
A = 0,1,0,1,...
B = 1,0,1,0,...
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