The desired result appears in M.H.A. Newman's classic book, *Elements of the Topology of Plane Sets of Points* (2nd ed., 1951), as Theorem 14.5 in Chapter VI, on pg. 164:

**Theorem 14.5:** *Every simple arc in $X^2$ is an arc of a simple closed curve in $X^2$.*

$X^2$ is Newman's notation for a space that is either the "open" plane, $R^2$, or the "closed" plane", $R^2 \cup \{\infty\}$.