Question about convergence in complex numbers field

Question

It may be a simple question, but if we want to show that $(z_n)\subset\mathbb{C}$ is convergent to $z\in\mathbb{C}$ then we should just check that absolute value of $z_n$ is convergent to absolute value of $z$ or we should check real part convergence and imaginary part convergence?

Answer 1

No. You should check that

$$|z_n-z|\xrightarrow{n\to\infty}0$$ which's equivalent to the convergence of the real and the imaginary part of $(z_n)$ and notice that we may have $|z_n|\xrightarrow{n\to\infty}|z|$ but $(z_n)$ isn't even convergent: take for example $z_n=(-1)^n$.

Answer 2

It's certainly not enough to check that $|z_n|\to |z|$. This doesn't even work in $\mathbb{R}$, since there are two separated numbers with any nonzero absolute value. It's even worse in $\mathbb{C}$, where there's a whole circle's worth of numbers of a given absolute value. It is possible to check that $|z_n-z|\to 0$. This might look similar, but it works, in a sense because $0$ is the only complex number with absolute value zero. A more careful argument uses continuity of the square and square root functions to show this convergence is equivalent to the convergence of each of the terms in the absolute value, which is precisely convergence in real and imaginary parts.