If E is a set and Y a point that is the limit of two sequences, {x_n} and {y_n} such that x_n is in E and y_n is an upper bound for E, prove that y = sup E. Is the converse true?

Let R be the set of all infinite sequences of real numbers, with the operations u+v=(u1,u2,u3,......)+(v1,v2,v3,......)=(u1+v1,u2+v2,u3+v3,.....) and cu=c(u1,u2,u3,......)=(cu1,cu2,cu3,......). Determine whether R is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.