  Finding independent sets of maximum size in fixed graphs is well known to be
an NP-hard task. Using scaling limits, we characterise the asymptotics of
sequential degree-greedy explorations and provide sufficient conditions for
this algorithm to find an independent set of asymptotically optimal size in
large sparse random graphs with given degree sequences. In the special case of
sparse Erd\"os-R\'enyi graphs, our results allow to give a simple proof of the
so-called $e$-phenomenon identified by Karp and Sipser for matchings and to
give an alternative characterisation of the asymptotic independence number.
