Suppose $v_{1}, v_{2}, v_{3}, v_{4}$ spans $V$. Prove that the list

also spans $V$.

We have $v_4$, and $v_3 = (v_3 - v_4) + v_4$ so we can also reach $v_3$ from our vectors, continue like this: $v_2 = (v_2 - v_3) + v_3$ and $v_1 = (v_1 - v_2) + v_2$
Now, any $v \in V$ can be written $v = a_1 v_1 + \dots + a_4v_4$ since $(v_1, \dots v_4)$ spans $V$, substitute the v's in terms of our new list to get $v = b_1 (v_1 - v_2) + b_2(v_2-v_3) + b_3(v_3-v_4) + b_4$.