Set of convergent sequences as a vector space, show associativity I try to prove that a set of convergent sequences $S=\{(a_n) : (a_n) \text{ convergent sequence of natural numbers} \}$
is a vector space.
I'm guess I have to show that all 8 axioms are valid, but I have a problem how to show the associativity. Should I do it with the epsilon criterion or is enough to say that the + operation on sequences behaves just like it would on the set of real numbers. 
 A: It might be more natual to show that the set of all real sequences is a real vector space -- this vector space really is like $\mathbb{R}^n$ only with coordinates indexed by all positive integes instead of just $\{1,\ldots,n\}$ -- and then show that the convergent sequences form a subspace.  In this case the calculus facts that you need to use are very familiar: e.g. that a sum of two convergent sequences is convergent.
At least this approach makes clear that you will not have to use any "epsilontics" to show the associativity of addtion.
A: To argue for associativity of addition, your second proposal is correct - there is nothing special about convergence here, and no limit-related argument is necessary. Addition of three non-convergent sequences $(x_n)$, $(y_n)$, and $(z_n)$ is associative for exactly the same reason.
In contrast, showing closure under the operations of addition and scalar multiplication requires a (comparatively) non-trivial amount of work. It is possible to add two non-convergent sequences together and get a convergent sequence, so we see that something about convergence must be used.
