  The primary purpose of this article is to show that a certain natural set of
axioms yields a completeness result for continuous first-order logic. In
particular, we show that in continuous first-order logic a set of formulae is
(completely) satisfiable if (and only if) it is consistent. From this result it
follows that continuous first-order logic also satisfies an \emph{approximated}
form of strong completeness, whereby $\Sigma\vDash\varphi$ (if and) only if
$\Sigma\vdash\varphi\dotminus 2^{-n}$ for all $n<\omega$. This approximated
form of strong completeness asserts that if $\Sigma\vDash\varphi$, then proofs
from $\Sigma$, being finite, can provide arbitrary better approximations of the
truth of $\varphi$.
