Existence of a complex sequence with given property

Question

How to show the existence of a complex sequence $(z_n)$ with $z_nne 1, forall n$ but $lim_{ntoinfty}z_n=1$ such that $lim_{nto infty}sin(frac{1}{1-z_n})=100$? Can such a sequence be explicitly written?

Kavi Rama Murthy · Answer

Hint: Solve $w^{2}-200iw-1=0$. Then choose $zeta_n to infty$ such that $e^{izeta_n} =w$ for all $n$. ( Take $c$ with $e^{ic}=w$ and take $zeta_n =2npi +c$). Finally take $z_n=1-frac  1{zeta_n}$.

Existence of a complex sequence with given property

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