  It is known that supercyclicity implies strong stability. It is not known
whether weak l-sequential supercyclicity implies weak stability. In this paper
we prove that weak l-sequential supercyclicity implies weak quasistability.
Corollaries concerning the characterisation of (i) weakly l-sequentially
supercyclic vectors that are not (strongly) supercyclic, and (ii) weakly
l-sequentially supercyclic isometries, are also proved.
