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Is it possible to construct a holomorphic function on $D(0,1)$ where $D$ is the disk with center $0$, and radius $1$, such that $f(\frac{1}{n}) = z_n$ when $z_n = 0$ for even $n$ and $z_n$ = $\frac{1}{n}$ for odd $n$

Attempt of solution: I have written down that since there is a convergent sequence of points to $0$ in $\frac{1}{n}$ so then $f = 0$ everywhere on $D.$ Does this imply that there is no such holomorphic function because for $n>0$, the real axis is a convergent sequence for $\frac{1}{n}$?

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I am not sure what you mean by

the real axis is a convergent sequence for $\frac{1}{n}$

but, indeed there is no such holomorphic $f$. Knowing that $f(\frac{1}{2n})=0$ for all $n$, the function has to be identically zero in $D(0,1)$. This contradicts the other requirements $f(\frac{1}{2n+1})=\frac{1}{2n+1}$.

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