I am trying to understand exercise 6.3.M(a) in Vakil's algebraic geometry notes. It goes as follows: Suppose $B$ is a ring. If $X$ is a $B$-scheme, and $f_0,...,f_n$ are $n+1$ functions on $X$ with no common zeros, then show that $[f_0,...,f_n]$ gives a morphism of $B$-schemes.
My attempt is as follows: Let $X_{f_i}$ denote the locus where $f_i$ doesn't vanish. We know from an earlier exercise that $X_{f_i}$ is open in $X$. For any affine open $V=\operatorname{Spec}A$ of $X$, we have $V_f:=X_{f_i}\cap V=D(f_i|_{V})$. So we can define a map $\varphi_i|_V:V_{f_i}\rightarrow U_i\subseteq\mathbb{P}_B^n$ by the morphism corresponding to the morphism of $B$-algebras $B[x_{0/i},...,x_{n/i}]$ given by $x_{j/i}=f_j/f_i$ (and here $x_{j/i}$ is notation for $x_j/x_i$).
My question is, where do I go from here? And is this the right approach? This question is labelled as "easy" and most of such labelled exercises in Vakil are pretty short, so I'm not sure if I'm on the best track.