Can any Continuous Process be physically realized ?

For an apple to grow full, it must first grow half. To grow half, it must first grow 1/4. To grow 1/4, it must first grow 1/8. And the series can be carried on ad infinitum. This series shows that an apple must go through infinite steps to grow full. Can we then conclude, on account of the infinite in our argument, that an apple can never grow into a full ruddy fruit?

The above is clearly an alternate statement of Zeno's paradox. The paradox as stated by Zeno or me or any other version is equivalent to stating the following:
        "No process is possible if it can be broken into an infinite series of physically realizable states."

Basically, we are grappling with infinity. The problem arises from the intuitive idea that an infinite number of steps is not physically realizable, at least not in a finite duration.

This raises some important questions. First, whether the above is a problem at all. That is, whether the splitting of a finite magnitude of something into an infinite number of magnitudes (most of which are infinitesimal) should be regarded as an obstruction to the realization of the whole magnitude if the process of such realization entails realization of each of the component magnitudes.

Secondly, if the realization of infinite states indeed poses trouble then whether each of the infinite states are physically realizable at all. That is, whether the attribute whose magnitude is being split/divided is continuous or discrete or something?

In other words, to solve the problem stated above one must prove either of the following two:

    1.      that an infinite number of states in a process of finite duration poses no problem at all; or

    2.      that, physically, no process of finite duration can go through an infinite number of states. (i.e. processes occur in a discrete fashion)

Before dealing with this problem properly, it would be useful to develop a better understanding of the more elementary concepts. This we shall pursue hereafter.