You can still have an expanding universe even if it is spatially infinite.
It just means that distant objects move apart, slowly at first, and then
faster and faster.
Suppose space were not expanding. It is three dimensional, obviously
(ignoring the "time dimension"). Would you demand that there exist a
four dimensional space in which the three dimensional one was embedded,
in order to give it reality? And would that four dimensional space have
to be part of a larger five dimensional space, which is itself part of
a six dimensional one, and so on?
All this is conceivable, but is it really necessary? I don't think we need
to assume there are higher dimensional spaces in order to give ours
reality.
Even though our space is said to be "curved", and expanding, those
properties do not require higher dimensional spaces, either. Curvature is
like a gradual internal warping of space, so that large enough triangles
don't have angles that add to exactly 180 degrees. It can be fully
described without reference to any larger embedding space.
Likewise the expansion of space can be expressed soley in terms of the
relations between objects within our universe. There is no need to
postulate a higher dimensional universe which ours is part of.
Hal