  Before we turn to the details, let us consider the parallel matter of sequences construed as functions on the natural numbers. Anyone who teaches about (infinite) sequences of some sort, say sequences of integers or real numbers, may at some point need to say what a sequence actually is.  Surely this is not done very often in elementary presentations: usually one would give examples instead of a formal definition, or illustrate what sequences are for by using them in some way or other.  In any case, it happens that in the usual set-theoretic modeling of mathematics, sequences of real numbers would be taken to be functions from the set of natural numbers to the set of real numbers.  So we have a reduction of one kind of object, sequences, to another, functions.  Of course, functions are then reduced to sets of ordered pairs, ordered pairs to sets of a certain form, natural numbers to sets of yet another form, and real numbers in their own way.  Concerning this kind of reduction, we should always ask whether it is necessary or silly, and whether it is useful to those using the mathematical objects in the first place. All of this is worth keeping in mind as we turn back to the sequences.