My Math Forum Limsup question

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 February 12th, 2017, 05:10 AM #1 Member   Joined: Jan 2016 From: Blackpool Posts: 95 Thanks: 2 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
 February 12th, 2017, 07:05 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 1,941 Thanks: 1008 let $a_n = \sin\left(\dfrac{2 \pi n}{N}\right)$ $b_n = \cos\left(\dfrac{2 \pi n}{N}\right)$ $\limsup\{a_n\} = 1$ $\limsup\{b_n\} = 1$ $\limsup\{a_n + b_n\} = \sqrt{2} \neq 2 = 1+1$ Thanks from Jaket1
 February 12th, 2017, 07:53 AM #3 Member   Joined: Jan 2016 From: Blackpool Posts: 95 Thanks: 2 Hi what is the difference between the "n" and the capital "N" thanks
 February 12th, 2017, 07:58 AM #4 Member   Joined: Jan 2016 From: Blackpool Posts: 95 Thanks: 2 I understand that An=cos(n) and Bn=sin(n) would also work
February 12th, 2017, 08:00 AM   #5
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 Originally Posted by Jaket1 Hi what is the difference between the "n" and the capital "N" thanks
$n$ is the index, $N$ is just some constant so you can sample the trig functions as finely as you like.

 February 13th, 2017, 05:28 AM #6 Senior Member   Joined: Feb 2012 Posts: 144 Thanks: 16 A simpler example: A = 0,1,0,1,... B = 1,0,1,0,... Thanks from Jaket1

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