= Solution
\b[No such <entire function> exists.] If it did, the entire function $h(z)^2-z$ would vanish at the distinct points $(n/(n+1))^2$, whose <accumulation point> is one. The <identity theorem> would give $h(z)^2=z$ on all of $\mathbb C$.
At zero this forces $h(0)=0$. Differentiating the identity gives $2h(z)h'(z)=1$, impossible at zero. Equivalently, a squared holomorphic function has an even-order zero, whereas $z$ has a simple zero. This is the obstruction expressed by <no entire square root of the identity function>; a square-root branch near one cannot extend holomorphically through zero.
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