  We consider a sequence of bubble converging minimal hypersurfaces, or H-CMC
hypersurfaces, in compact Riemannian manifolds without boundary, of dimension
4, 5, 6 or 7, and prove upper semicontinuity of index plus nullity, for such a
bubble converging sequence. This complements the previously known lower
semicontinuity of index obtained by Buzano--Sharp, and Bourni--Sharp--Tinaglia.
  The strategy of our proof is to analyse a weighted eigenvalue problem along
our sequence of hypersurfaces. This strategy is inspired by the recent work of
Da Lio--Gianocca--Rivi\`{e}re. A key aspect of our proof is making use of a
Lorentz--Sobolev inequality to study the behaviour of these weighted
eigenfunctions on the neck regions along the sequence, as well as the index and
nullity of our non-compact bubbles.
