We present three constructions of complete caps in PG(d,q), q odd, where complete caps in a projective space of smaller dimension are involved. We thereby obtain new series of upper bounds on n 2 (d,q), the smallest number of points in a complete cap in PG(d,q). The constructions show that for k>=0, n 2 (k+1,3)=<2n 2 (k,3); n 2 (4k+2,q)=<q 2 k + 1 +n 2 (2k,q) for q>=5 an odd prime power; and n 2 (4k+2,q)=<q 2 k + 1 -(q+1)+n 2 (2k,q)+n 2 (2,q) for q>=9 an odd prime power.