February 12th, 2017, 06:10 AM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 100 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, 08:05 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,172 Thanks: 1142 
let $a_n = \sin\left(\dfrac{2 \pi n}{N}\right)$ $b_n = \cos\left(\dfrac{2 \pi n}{N}\right)$ $\limsup\{a_n\} = 1$ $\limsup\{b_n\} = 1$ $\limsup\{a_n + b_n\} = \sqrt{2} \neq 2 = 1+1$ 
February 12th, 2017, 08:53 AM  #3 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 100 Thanks: 2 
Hi what is the difference between the "n" and the capital "N" thanks 
February 12th, 2017, 08:58 AM  #4 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 100 Thanks: 2 
I understand that An=cos(n) and Bn=sin(n) would also work 
February 12th, 2017, 09:00 AM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 2,172 Thanks: 1142  
February 13th, 2017, 06:28 AM  #6 
Senior Member Joined: Feb 2012 Posts: 144 Thanks: 16 
A simpler example: A = 0,1,0,1,... B = 1,0,1,0,... 

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