Rearrangement in proof for Euler's formula

Rearrangements are not the issue here. The basic result is this:

Thm: Suppose $z_n=x_n+iy_n$ is a sequence of complex numbers, where $x_n,y_n$ are the real and imaginary parts of $z_n.$ Then $\sum_{n=1}^{\infty} z_n$ converges in $\mathbb C$ iff both $\sum_{n=1}^{\infty} x_n$ and $\sum_{n=1}^{\infty} y_n$ converge in $\mathbb R.$ In the case of convergence we have

$$\sum_{n=1}^{\infty} z_n = \left(\sum_{n=1}^{\infty} x_n\right )+i\left(\sum_{n=1}^{\infty} y_n\right).$$

This implies Euler's formula for $e^{ix}.$

The complex numbers work the same way $\mathbb{R}^2$ does as a metric spaces but the generalization is the same for $\mathbb{R}^n$, we pass from the point to its length by using the distance function induced by the Euclidean norm. This often requires repeated use of the triangle inequality but the flow of the argument remains essentially the same for the multivariable Taylor series.