How can one show that the [Sorgenfrey line](http://www.proofwiki.org/wiki/Definition:Sorgenfrey_Line) is hereditarily Lindelöf (that is, all subspaces of the Sorgenfrey line are Lindelöf)? 

I know the Sorgenfrey line is Lindelöf and hence every closed subspace is Lindelöf.