  In this paper, we prove the existence of a spherical $t$-design formed by
adding extra points to an arbitrarily given point set on the sphere and,
subsequently, deduce the existence of nested spherical designs. Estimates on
the number of required points are also given. For the case that the given point
set is a spherical $t_1$-design such that $t_1 < t$ and the number of points is
of optimal order $t_1^d$, we show that the upper bound of the total number of
extra points and given points for forming nested spherical $t$-design is of
order $t^{2d+1}$. A brief discussion concerning the optimal order in nested
spherical designs is also given.
