Proving convergence of real and imaginary parts

I am trying to prove that a complex sequence $(z_n)$ converges if and only if $(\Re(z_n))$ and $(\Im(z_n))$ converge. Now $\impliedby$ was straightforward, but I got a bit stuck with $\implies$:

$$|z_n-z| = |(x_n+iy_n)-(x+iy)|$$ $$=|(x_n-x)-i(y_n-y)|$$ $$\geq||x_n-x|-|y_n-y||$$

Now $\displaystyle\lim_{n\to\infty} z_n =0 \implies \displaystyle\lim_{n\to\infty} ||x_n-x|-|y_n-y|| = 0$

Which then leaves me with having $\displaystyle\lim_{n\to\infty} |x_n-x|=\displaystyle\lim_{n\to\infty} |y_n-y|$, and unsure what to do to deduce these limits are zero.

How should I proceed, or how should I begin this half of the proof if this is a dead end?

Solution 1:

Use the fact that for any $z\in\mathbb{C}$, you have $$ |z| \ge |\Im(z)|$$ and $$|z| \ge |\Re(z)|.$$