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I have an upper bound, say α = 0+di. My question is does only the real part, Re(α) = 0 define the upper bound? Or does the Im(α) = d have nothing to do with bounds in general?

Since it seems to me if I have the lexicographic ordering on ℂ such as for any two m,n [itex]\in[/itex] ℂ, where m = a+bi and n = c+di and I define the ordering as m<n if a<c or if a=c and b<d.

The last bit, b<d gives me the impression that Im(α) would play a role in the upper bound. The reason I am asking is because in a proof I read, they prove this order has no least upper bound as there are infinitely many complex numbers with their real parts equal to Re(α) but different imaginary parts. So, I guess if only the real parts of complex numbers define the bounds then it makes sense to me.