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.
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poles are isolated singularities, so they must form a discrete (and thus finite, as the sphere is compact) subset – user8268 Oct 03 '20 at 13:37
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That's right,but I want to know the proof by identity theorem because in forster's book conclude this by identity theorem but didn't explain how.@user8268 – Mathgreek Oct 03 '20 at 13:46
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then look at the zeroes of $1/f$ (which is basically what you wrote) – user8268 Oct 03 '20 at 16:23
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@user8268.so, is this a correct solution? – Mathgreek Oct 03 '20 at 16:46