Poles of a meromorphic function on riemann sphere is finite

Question

By identity theorem,I want to prove that every meromorphic function $f$ on riemann sphere $\mathbb{P}^1$ has finitely many poles, can this solution be true? : Suppose $f$ has infinite poles then the restriction $f_{|\mathbb{P}^1\backslash\text{poles}}$ of $f$ is holomorphic and this set has limit point in $\mathbb{P}^1$ and since $f$ is continuous, $\lim f=f$ on this restricted set. Because this limit is infinite we conclude that $f$ is infinite every where and it's a contradiction.