A clean way to separate sets of surreals

Abstract

Let surreal numbers be defined by means of sign sequences. We give a proof that if $S < T$ are sets of surreals, then there is some surreal $w$ such that $S < w < T$. The classical proof is simplified by observing that, for every set $S$ of surreals, there exists a surreal $s$ such that, for every surreal $w$, we have $S<w$ if and only if the restriction of $w$ to the length of $s$ is $ \geq s$. Hence $S < w < T$ if and only if $w$ satisfies the above condition, as well as its symmetrical version with respect to $T$. It is now enough to check that if $S < T$, then the two conditions are compatible.