Finiteness in Prime Decomposition Theorem for 3 manifolds

Question

In Allen Hatcher's text on 3 manifolds, he proves the Prime Decomposition Theorem by showing that a collection S of 2 spheres embedded in a smooth and compact 3-manifold M satisfying the condition that no component of M|S is a punctured 3 ball necessarily has a bound on the number of spheres in S.
To show this, he takes a triangulation of M, which will necessarily have finitely many 3 simplices. Then he makes the collection S "transverse to the simplices" in the following way: "First make S disjoint from vertices, then transverse to edges, meeting them in finitely many points, then transverse to 2 simplices, meeting them in finitely many arcs and circles."
My question is, since we do not initially know that the collection S is finite, how can we ensure that S meets the simplices in finitely many points or arcs? Intuitively it seems to me that if it meets infinitely many times then the spheres must be nested, which will perhaps mean the component between them is a punctured 3 ball, but is this necessarily the case?

Answer 1

The standard proof of finiteness comes with a uniform upper bound $s$ on the number of pairwise-nonparallel spheres. In order to prove this bound you consider a collection of $s+1$ pairwise non-parallel spheres and get a contradiction. From this, the finiteness result is immediate.