Let $f,g$ be two homogenous polynomial of degree $d$ in $\Bbb{C}[x_1,...,x_{n+1}]$. define the quotient $f/g$ on $\Bbb{CP}^n$, intuitively it should be the meromorphic function, but I want to prove it rigorously.
That is on each point, the meromorphic function is the quotient of two elements on $\mathcal{O}_{\Bbb{CP}^n,x}$. The problem is $f,g$ is not defined as holomorphic function on $\Bbb{CP}^n$. We can not check it using the definition directly.
View elements of $\Bbb{P}^n$ in homogeneous coordinates as $[x_0:\cdots:x_n]$. We can evaluate a ratio of degree $d$ polynomials on such data by $(f/g)([x_0:\cdots:x_n])=f(x_0,\ldots,x_n)/g(x_0,\ldots, x_n)$. This is well-defined because $$f(\lambda x_0,\ldots, \lambda x_n)/g(\lambda x_0,\ldots, \lambda x_n) = \lambda^d f(x_0,\ldots, x_n)/\lambda^d g(x_0,\ldots, x_n)= f(x_0,\ldots,x_n)/g(x_0,\ldots, x_n).$$ So, you have a globally defined function (at least where $g(x_0,\ldots, x_n)\ne0$). Now, use the standard charts on $\Bbb{P}^n$ to write this ratio of functions as a function on $\mathbb{A}^n$ with coordinates $(x_0/x_i,\ldots, 1,\ldots, x_n/x_i)$.