Are pseudocompact metric spaces complete?

Question

Is there a way to show that pseudocompactness on a metric space implies completeness directly (without using sequential compactness)?

Answer 1

Suppose that $X$ is a metric space, and that $\langle x_n:n\in\Bbb N\rangle$ is a Cauchy sequence in $X$ that does not converge. Without loss of generality we may assume that $x_m\ne x_n$ whenever $m,n\in\Bbb N$ and $m\ne n$. Let $D=\{x_n:n\in\Bbb N\}$; then $D$ is a closed discrete set in $X$, so the map $f:D\to\Bbb R:x_n\mapsto n$ is continuous. By the Tietze extension theorem $f$ extends to a continuous $F:X\to\Bbb R$, so $X$ is not pseudocompact.