For $j=0,\ldots,n$ consider the affine hyperplane $A_j:=e_j+\langle e_0,\ldots,e_{j-1},e_{j+1},\ldots,e_n\rangle$ in $\mathbb K^{n+1}$ and the associated embedding $\tau_j:\mathbb K^n\rightarrow\mathbb KP^n, \tau_j(x_1,\ldots,x_n):=[x_1:\ldots:x_j:1:x_{j+1}:\ldots:x_n]$, where $e_j\in\mathbb K^{n+1}$ the $j'$th unit vector is.
Now I come to my question; How can I show that the images of $\tau_j$ overlay whole $\mathbb KP^n$ or mathematically spoken: $\mathbb KP^n=\bigcup_{j=0}^n \tau_j(\mathbb K^n)$
I think it should be a short proof but since I am a newbie in projective geometry I do not really have an idea.
Given $[x_0:\ldots:x_n]$, there exists $j$ such that $x_j\neq0$; therefore $$\left[\frac{x_0}{x_j}:\ldots:\frac{x_n}{x_j}\right]=[x_0:\ldots:x_n]$$ has a $1$ in the $(j+1)$-th place, therefore sits in the image of $\tau_{j}$. As the point was generic, this shows that the images of these maps cover the whole projective space.