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sequences

if you have a sequence ana_n and a sequence bnb_n, is there such a sequence an+bna_n+b_n? I think there is, but i am not sure
Original post by asdfyolo
if you have a sequence ana_n and a sequence bnb_n, is there such a sequence an+bna_n+b_n? I think there is, but i am not sure


You can certainly create such a sequence by adding corresponding terms, or you could multiply them to get a sequence (anbn)(a_nb_n), or take any function of the two (f(an,bn))(f(a_n,b_n)) as long as (an) and (bn)(a_n)\text{ and }(b_n) are in the domain of f.
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asdfyolo
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Original post by ghostwalker
You can certainly create such a sequence by adding corresponding terms, or you could multiply them to get a sequence (anbn)(a_nb_n), or take any function of the two (f(an,bn))(f(a_n,b_n)) as long as (an) and (bn)(a_n)\text{ and }(b_n) are in the domain of f.

thanks. does that have anything to do with answering this: suppose an≤bna_n \leq b_n and the limit of the first sequence is a, and the limit of the second sequence is b, show a≤ba \leq b.i have been told it has something to do with showing a−b≤za-b \leq z for all z>0
Original post by asdfyolo
thanks. does that have anything to do with answering this: suppose an≤bna_n \leq b_n and the limit of the first sequence is a, and the limit of the second sequence is b, show a≤ba \leq b.i have been told it has something to do with showing a−b≤za-b \leq z for all z>0


Told by whom. Seems an odd inequality, when you could just show a-b <= 0. Not sure how they're envisioning you do it with going for that inequality.

Also going to depend on what you've covered already regarding the limit of the sum/difference of two sequences.

And I'm a bit puzzled why you were asking about a_n + b_n, since that doesn't come into it.


I'd go for a proof by contradiction myself.
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asdfyolo
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Original post by ghostwalker
Told by whom. Seems an odd inequality, when you could just show a-b <= 0. Not sure how they're envisioning you do it with going for that inequality.

Also going to depend on what you've covered already regarding the limit of the sum/difference of two sequences.

And I'm a bit puzzled why you were asking about a_n + b_n, since that doesn't come into it.


I'd go for a proof by contradiction myself.


told by a lecturer, we have covered the limit of the sum of two sequences (i just found out now), i have to do it that weird way
Original post by asdfyolo
told by a lecturer, we have covered the limit of the sum of two sequences (i just found out now), i have to do it that weird way


OK.

Now that I look at it again, it's not that bad.

If we let bn−an=cnb_n-a_n=c_n which is >= 0.

We have bn=an+cnb_n=a_n +c_n

and now consider the limits.

I presume you can assume the limit of a positive sequence is >= 0.

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