# Question: consider infinite sequences a0 a1 a2 where aj ...

###### Question details

Consider infinite sequences a_{0}, a_{1},
a_{2}, ... where a_{j} ∈ **R**; we
will denote such sequences as {a_{j}}. Define addition of
two sequences {a_{j}} and {b_{j}} by the sequence
a_{0}+b_{0}, a_{1}+b_{1}, ... and
scalar multiplication by r ∈ R as ra_{0}, ra_{1},
ra_{2}, .... It can be then shown that the set of all
infinite sequences over the field **R** form a Vector
Space (you do not have to show this).

(a) Let S be the subset of sequences that satisfy a_{n}
= a_{n−1} + a_{n−2} for all n ≥ 2. Prove that S is
also a Vector Space.

(b) Let {f_{n}} be the Fibonacci sequence 1, 1, 2, 3, 5,
8, .... Note that {f_{n}} ∈ S. Find another sequence
{c_{n}} ∈ S such that the two sequences {{f_{n}},
{cn}} form a basis for S.