Limit of Hilger real and imag parts yields classical

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Theorem: The following formula holds: $$\displaystyle\lim_{h \rightarrow 0^+} \left[ \mathrm{Re}_h(z) + \mathring{\iota} \mathrm{Im}_h(z) \right] = \mathrm{Re}(z) + i \mathrm{Im}(z).$$

Proof: