Proving the Limit of a Sum using Sequence Convergence

[marlen]

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. :)

 

[EnumaElish]

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.

 

[SiddharthM]

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

 

[HallsofIvy]

You will also want to use $|a+ b|\le |a|+ |b|$.