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. 

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.
