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  "body": "\n## Proof\n\nWe argue by contradiction.\n\n1. Let $c$, $r$ and $f$ be as in the statement, with $c$ an [[essential singularity|0YKDJBV26S]] of $f$. Fix $\\delta \\in (0, r)$ and let $P = \\set{z : 0 \u003c \\abs{z - c} \u003c \\delta}$.\n2. Assume that $f(P)$ is not dense in $\\C$. Then there are $w_0 \\in \\C$ and $\\varepsilon \u003e 0$ with $\\abs{f(z) - w_0} \\ge \\varepsilon$ for each $z \\in P$.\n3. Let $g(z) = 1/(f(z) - w_0)$ on $P$. It is [[holomorphic|0DG0Z72CQD]] on $P$, as $f - w_0$ is holomorphic and nonzero there, and $\\abs{g} \\le 1/\\varepsilon$.\n4. By [[Riemann's theorem on removable singularities|0BSQDZWF34]], $g$ extends to a holomorphic function $G$ on $\\set{z : \\abs{z - c} \u003c \\delta}$. Also $G(z) \\ne 0$ for $z \\in P$.\n5. Case $G(c) \\ne 0$. By continuity, $G \\ne 0$ on a disc $\\abs{z - c} \u003c \\delta'$, so $w_0 + 1/G$ is holomorphic there. It equals $f$ for $0 \u003c \\abs{z - c} \u003c \\delta'$.\n6. Define $F = f$ on $0 \u003c \\abs{z - c} \u003c r$ and $F(c) = w_0 + 1/G(c)$. Near $c$, $F = w_0 + 1/G$, so $F$ is [[complex differentiable|0D1C9Y1EXA]] at $c$.\n7. So $F$ is holomorphic on $\\abs{z - c} \u003c r$ and equals $f$ off $c$. Thus $c$ is a [[removable singularity|0P18JC72RE]] of $f$, which contradicts step 1.\n8. Case $G(c) = 0$. By continuity, $G(z) \\to 0$ as $z \\to c$. Since $\\abs{f(z)} \\ge \\abs{f(z) - w_0} - \\abs{w_0} = 1/\\abs{G(z)} - \\abs{w_0}$ on $P$, we get $\\abs{f(z)} \\to \\infty$.\n9. So $c$ is a [[pole|0WPQYW8MTE]] of $f$, which also contradicts step 1.\n10. Both cases are impossible, so $f(P)$ is dense in $\\C$.\n11. Let $w \\in \\C$. For each $n \\ge 1$, apply step 10 with $\\delta_n = \\min(\\delta, 1/n)$. It gives $z_n$ with $0 \u003c \\abs{z_n - c} \u003c 1/n$ and $\\abs{f(z_n) - w} \u003c 1/n$.\n12. Then $z_n \\ne c$, $z_n \\to c$ and $f(z_n) \\to w$.\n\n## Idea\n\nIf $f$ stayed away from a value $w_0$ near $c$, then $1/(f - w_0)$ would be bounded near $c$. Riemann's theorem removes its singularity. Turning back to $f$, the singularity of $f$ is removable or a pole, depending on whether this extension vanishes at $c$.\n",
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