Is the quotient space finite-dimensional linear space?

Question

We have space $V=\{(x_1,x_2,x_3..):(x_1,2x_2, 3x_3,...)\in l_{\infty}\}$ Is the factor space (quotient-space) $l_{\infty}/V$ finite-dimensional linear space? First of all in this exercise (I think)we must prove that $V$ is a linear. Really need help/explanations with this exercise.

Answer 1

Hint: Note that $V\subset c_0$, so we have a surjective linear map $T:\ell_\infty/V\to \ell_\infty/c_0$ given by $T(x+V)=x+c_0$. Thus in order to show that $\ell_\infty/V$ is infinite-dimensional, it suffices to show that $\ell_\infty/c_0$ is infinite-dimensional.