  We study the red sequence in a cluster of galaxies at z=1.62 and follow its
evolution over the intervening 9.5 Gyr to the present day. Using deep YJKs
imaging with the HAWK-I instrument on the VLT we identify a tight red sequence
and construct its rest-frame i-band luminosity function (LF). There is a marked
deficit of faint red galaxies in the cluster that causes a turnover in the LF.
We compare the red sequence LF to that for clusters at z<0.8 correcting the
luminosities for passive evolution. The shape of the cluster red sequence LF
does not evolve between z=1.62 and z=0.6 but at z<0.6 the faint population
builds up significantly. Meanwhile, between z=1.62 to 0.6 the inferred total
light on the red sequence grows by a factor of about 2 and the bright end of
the LF becomes more populated. We construct a simple model for red sequence
evolution that grows the red sequence in total luminosity and matches the
constant LF shape at z>0.6. In this model the cluster accretes blue galaxies
from the field that are then quenched and subsequently allowed to merge. We
find that 3--4 mergers among cluster galaxies during the 4 Gyr between z=1.62
and z=0.6 matches the observed luminosity function evolution between the two
redshifts. The inferred merger rate is consistent with other studies of this
cluster. Our result supports the picture that galaxy merging during the major
growth phase of massive clusters is an important process in shaping the red
sequence population at all luminosities.
