The author constructs an approximating fixed point sequence

$\left\{{x}_{n}\right\}$ for a mapping

$T$ of a nonempty closed convex subset

$C$ of a smooth and uniformly convex Banach space

$X$, generated by the iteration process as the projection of an arbitrary initial point

${x}_{0}$ into the intersection of two closed convex subsets

${C}_{n}$,

${Q}_{n}$ of

$C$, given by

${x}_{n+1}={P}_{{C}_{n}\cap {Q}_{n}}{x}_{n}$. The author goes on to prove that the sequence generated is an approximating fixed point sequence for

$T:C\to C$ and that it is strongly convergent to a fixed point of

$T$.