I am reading Rick's Miranda book and he's now talking about how we can do a laurent series approximation in an Algebraic curve,page $173$, that is
Suppose that $X$ is an algebraic curve, fix a finite number of points $p_1,...,p_n$ in $X$, choose a local coordinate $z_i$ at $p_i$ and choose laurent polynomials $r_i(z_i)$ for each $i$. Then there is a global meromorphic function $f$ on $X$ such that for every $i$ f has $r_i$ has a laurent tail at $p$.
To prove this result he uses this lemma
Let $X$ be an algebraic curve, then for any finite number of points $p,q_1,...,q_n$ in $X$ and any $N\geq1$ there is a global meromorphic function $f$ on $X$ with $ord_{p}(f-1)\geq N$ and $ord_{q_i}(f)\geq N$.
Now what's confusing me its $f-1$, is $1$ supposed to be the constant function equal to $1$ and I don't see why this will he useful when he uses this fact in the proof, so any enlightenment would be helpull. Also a thing thats confusing is that he adds zeroes to $r_i$ so latter I cant tell if he is working with the original $r_i$ or this addeed zeroes $r_i$.
To prove the result we states fix an integer $N$ larger then every exponent of every term of every $r_i$. Extend each $r_i$ be adding zerot terms,and consider each $r_i$ as a laurent series with terms of degree less than $N$.For $f$ to have $r_i$ as its Laurent tail we need that $ord_{p_i}(f-r_i)\geq N$. By a previous lemma we know that there exist functions $g_i$ such that each $g_i$ as $r_i$ as a laurent tail at $p_i$. Let $M=min({ord_{p_i}(r_i)})$, by another previous lemma we know that there exist functions $h_i$ such that $ord_{p_i}(h_i-1)\geq N-M$ and $ord_{p_j}(h_i)\geq N-M$. Consider the function $f=\sum_ih_ig_i$, it will be the desired function.
Now I cant really see why this is the desired function.
I guess this is more of a question about laurent series because I just cant see that the funcion will have the desired laurent tails, so any enlightment would be nice.
Thanks in advance.