'Identity theorem' for Meromorphic functions If $f_1,f_2$ are meromorphic functions in $D$ and there exists a sequence of pairwise distinct points $z_n \in D$ such that $z_n \to z_o \in D$ and $f_1(z_n)=f_2(z_n),$ then $f_{1} \equiv f_2$ on $D.$ 
Appreciate if someone could advise me on how to prove this. Thank you. 

Since $f_1$ and $f_2$ are meromorphic in $D$ then there exists a holomorphic functions $a,b,c,d$ in $D$ such that $f_1=a/b$ and $f_2=c/d$, now let $h=ad-bc$ it is a holomorphic function on $D$, and for all $z_k$ we have $h(z_k)=0$, by continuity of $h$ and the fact that $z_k\to z_0$, we get $h(z_0)=0$, hence $h=0$, so that $a/b=c/d$ i.e $f_1=f_2$.
