  We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$
representations of the mapping class groups using peak sections in Kodaira
embedding. We give also estimates on the norm of the parallell transport of the
projective connection on the Verlinde bundle. The faithfulness has been proved
earlier in [1] using Toeplitz operators of compact K\"ahler manifolds and in
[10] using skein theory.
