My professor gave the Theorem: If $L/F$ and $F/E$ are (not necessarily finite) extensions, then $L/E$ is separable if and only if $L/F$ is separable and $F/E$ is separable. He proved that: If $L/F$ and $F/E$ are separable, then $L/E$ is separable. And said that the other if then statement is "obvious", i.e. If $L/E$ is separable then $L/F$ and $F/E$ are separable. But I just don't see how this is "obvious", I can't really prove it either.
Any help for this is appreciated.