  We use the compactness theorem of continuous logic to give a new proof that
$L^r([0,1]; \mathbb{R})$ isometrically embeds into $L^p([0,1]; \mathbb{R})$
whenever $1 \leq p \leq r \leq 2$. We will also give a proof for the complex
case. This will involve a new characterization of complex $L^p$ spaces based on
Banach lattices.
