Let V be the set of all infinite sequences (a0, a1, a2,...) of real numbers. Define addition

Question:

Let V be the set of all infinite sequences
(a0, a1, a2,...) of real numbers. Define addition and scalar multiplication by (a0, a1,...) + (b0, b1,...) = (a0 + b0, a1 + b1,...) and r(a0, a1,...) = (ra0, ra1,...). and r(a0, a1,...) = (ra0, ra1,...).
(a) Show that V is a vector space.
(b) Show that V is not finite dimensional.