A basic lemma about finite abelian group

Maybe I don't understand the proof of this lemma (from LAng, Algebra).
Please, help me.

LEMMA: Let G be a finite abelian group. Suppose that G has exponent n.
Then the order of G divides some power of n.

Proof: (by induction) Let b in G, b not= 1 and let H be the cyclic
subgroup generated by b. Then the order of H divides n since b^n=1,
and n is an exponent for G/H. Hence the order of G/H divides a power
of n by induction .....

Comments: We want to prove that if G has exponent n, then the order
of G divides some power of n. The last sentence says that, since G/H
has exponent n, then the order of G/H divides a power of n. But you
cannot use the thesis in the proof!

.

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