  We show that for the generic sequence (a) of elements in a subset A of a
separable locally convex metrisable space V, the sequences [T^k(a)]_n,
n=1,2,... are dense in the convex hull convA of A for all k=1,2,...; where T is
the Cesaro operator. Further, if convA is dense in V, then for every sequence
x_k of elements of V, k=1,2,... there exists a sequence of indices l_n,
n=1,2,..., such that, we have the simultaneous approximation that
[T^k(a)]_(l_n) converges to x_k as n tends to infinity for all k=1,2,... These
phenomena are topologically generic and in the case where A is a vector space
they are algebraically generic.
