Best constants for Lipschitz embeddings of metric spaces into c(0)

We answer a question of Aharoni by showing that every separable metric space can be Lipschitz 2-embedded into c(0) and this result is sharp; this improves earlier estimates of Aharoni, Assouad and Pelant. We use our methods to examine the best constant for Lipschitz embeddings of the classical l(p)-spaces into c(0) and give other applications. We prove that if a Banach space embeds almost isometrically into c(0), then it embeds linearly almost isometrically into c(0). We also study Lipschitz embeddings into c(0)(+).