= Solution
The <identity theorem> says that two holomorphic functions on a connected domain which agree on a set with an accumulation point inside that domain agree everywhere there.
\b[No such <entire function> exists.] On the <subsequence> $1/(2m)\to0$ indexed by <even numbers>, the proposed function equals $\sin z$. The <identity theorem> would force $f(z)=\sin z$ throughout $\mathbb C$. At the points indexed by <odd numbers> it is instead required to equal $-\sin z$, a contradiction since $\sin(1/n)>0$ for positive $n$. The accumulation point zero lies inside the assumed domain of holomorphy, which is essential to this argument.
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