  We verify Shalom's conjecture for the simple real-rank-one Lie group Sp(n ,1)
for any n: i.e. we show that it admits a metrically proper affine action on a
Hilbert space whose linear part is a uniformly bounded representation. We
provide two different proofs. Both approaches crucially use results on
uniformly bounded representations by Michael Cowling. The first approach is
quite abstract: it uses an automatic-properness result of Shalom and requires
almost no computations. The second approach is explicit: we deduce the
properness of cocycles from the non-continuity of a critical case of the
Sobolev embedding. This work is inspired from Pierre Julg's work on the
Baum--Connes conjecture for Sp(n,1).
