Consider the linear space $c^{(3)}$ of all sequences $x = (x_n)_{n=1}^{\infty}$ such that $\{x_{3k+q} \}_{k=0}^{\infty}$ converges for $q = 0,1,2$. Find the dimension and a basis for $c^{(3)}/c_0$. Note that $c_0$ is the linear space of sequences that converge to $0$.
I think the dimension is $1$ using the reasoning in the answer to this question. We know that any integer can be represented as $3k, 3k+1$ or $3k+2$. So $c^{(3)}$ is equivalent to $c$ (e.g. all convergent sequences)? Is this correct?