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Math Beauty and Her Limits 
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#1
Nov107, 08:00 PM

P: 16

Hello,
Can someone please help me prove that the lim as n goes to infinity of (the sequence an + the sequence bn) = (the lim of an) + (the lim of bn). What I think is that if one adds the two limits an + bn, she will come up with a new sequence cn and take its limit, which equals L. Then if she takes the limit of an and set it equal to L1 and take the limit of bn and set it equal to L2... After this I don't know. I don't even know if this makes sense. Someone please help me! I hope all of this makes sense. :) 


#2
Nov107, 09:17 PM

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P: 2,481

1. Since c_{n} > L, there must be a relationship between the elements of c_{n} and the number L. State that relationship.
2. Now you need to show that the relationship stated in "1" indeed holds. To show this: a. assume L = L1 + L2 b. use the given fact that the relationship stated in "1" holds between a_{n} and L1, as well as between b_{n} and L2, to show that when L = L1 + L2, the relationship in "1" holds between c_{n} and L. 


#3
Nov207, 12:44 PM

P: 176

you could use the epsilonN definition to show it too. it's a pretty straightforward application of the triangle inequality.



#4
Nov307, 06:35 AM

Math
Emeritus
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Thanks
PF Gold
P: 39,568

Math Beauty and Her Limits
You will also want to use [itex]a+ b\le a+ b[/itex].



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