MHB Problem about equivalent conditions for a basis of a free module

Ganitadnya
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Here's the problem:
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I don't see why there should be only finitely many nonzero a_z in b. I was able to prove uniqueness assuming that there only finitely many nonzero. I was able to show b implies d and b implies c, c implies a.
 

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So you're referring to the implication (a) $\implies$ (b). There are only finitely many $a_z$ in (b) simply because $Z$ is a generating set for $M$ (since it is a basis by (a)).
 

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