Chapter 4.2, Problem 48E

Let $R^{\infty }$ be the set of all infinite sequences of real numbers, with the operations

$\begin{array}{rll}u+v& =& \left(u_1,u_2,u_3,\mathrm{......}\right)+\left(v_1,v_2,v_3,\mathrm{......}\right)\\ & =& \left(u_1+v_1,u_2+v_2,u_3+v_3,\mathrm{.....}\right)\end{array}$

and

$\begin{array}{rll}cu& =& c\left(u_1,u_2,u_3,\mathrm{......}\right)\\ & =& \left(c{u}_1,c{u}_2,c{u}_3,\mathrm{......}\right).\end{array}$

Determine whether $R^{\infty }$ is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.