Prove that $\mathbb{C}$ with its usual complex absolute value $|\cdot |$ is complete.

Question

I know in order to prove that $\mathbb{C}$ is complete in need to show that its Cauchy Sequences converges, but I don't have an background in complex sequences so I don't know where to start regarding how to show they converge.

Answer 1

Assuming you know that $\mathbb{R}$ is complete, check that the following statements hold:

1. A sequence of complex numbers $(a_n + ib_n)_{n \in \mathbb{N}}$ with $a_n, b_n \in \mathbb{R}$ converges to $a + ib$ ($a, b \in \mathbb{R}$) if and only if $(a_n)_n \to a$ and $(b_n)_n \to b$ as real sequences.
2. A sequence of complex numbers $(a_n + ib_n)_{n \in \mathbb{N}}$ with $a_n, b_n \in \mathbb{R}$ is a Cauchy sequence if and only if $(a_n)_n$ and $(b_n)_n$ are real Cauchy sequences.

Note that the completeness of $\mathbb{C}$ now readily follows from $\mathbb{R}$'s completeness.