To convince yourself consider you have 8 exposures with sky RMS equal to 16 originally: 4 you combine, so the sky RMS becomes $\frac{1}{\sqrt{\frac{1}{16^2}+\frac{1}{16^2}+\frac{1}{16^2}+\frac{1}{16^2}}}=\frac{1}{\sqrt{\frac{4}{16^2}}}=8$. Now later on you want to combine the 5 images you have. Clearly the optimal weighting you can possibly get will be $\frac{1}{\sqrt{\frac{8}{16^2}}}=\frac{8}{\sqrt{2}}=5.65$ (the direct stack of the original 8 images). With our already stacked image added to our 4 others we can achieve the same S/N by weighting our stacks by the inverse variance: $\frac{1}{\sqrt{\frac{1}{16^2}+\frac{1}{16^2}+\frac{1}{16^2}+\frac{1}{16^2}+\frac{1}{8^2}}}=\frac{1}{\sqrt{\frac{4}{16^2}+\frac{1}{8^2}}}=\frac{1}{\sqrt{\frac{8}{16^2}}}=\frac{8}{\sqrt{2}}=5.65$.
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