Prove true for n = 3
z_3=(1+i)\left(1+\frac{i}{|1+i|}\right)
=(1+i)\left(1+\frac{i}{\sqrt{2}}\right)
=\frac{\sqrt{2}-1}{\sqrt{2}}+\left(\frac{1+\sqrt{2}}{\sqrt{2}}\right)i
|z_3|=\sqrt{\left(\frac{\sqrt{2}-1}{\sqrt{2}}\right)^2+\left(\frac{1+\sqrt{2}}{\sqrt{2}}\right)^2}
=\sqrt{3}
Assume true...