The Completeness of the Field of Complex Numbers

In general, a metric space has the property that every convergent sequence is Cauchy, but the converse, i.e., that every Cauchy sequence is convergent is NOT held by all metric spaces. When a metric space has the property that every Cauchy sequence converges in that space then the space is said to be complete. As we know, the set of real numbers is complete. The following theorem tells us that the set of complex numbers is also complete.

Theorem 1: The set of complex numbers $\mathbb{C}$ is complete, i.e., every Cauchy sequence in $\mathbb{C}$ converges in $\mathbb{C}$.

• Proof: Let $(z_n) = (x_n + iy_n)$ be a Cauchy sequence of complex numbers.

• From the A Sequence of Complex Numbers is Cauchy IFF The Real Part Sequence and Imaginary Part Sequence are Cauchy page we know that both $(x_n)$ and $(y_n)$ are Cauchy. But since $\mathbb{R}$ is complete, this implies that $(x_n)$ and $(y_n)$ both converge.

• From the A Sequence of Complex Numbers Converges IFF The Real Part Sequence and Imaginary Part Sequence Converges page we know that the convergence of $(x_n)$ and $(y_n)$ implies the convergence of $(z_n)$. $\blacksquare$