Suppose that {an} and {bn} are sequences of positive terms, and that the limit as n goes to infinity of (an/bn) = L > 0. Prove that limit as n goes to infinity of an is positive infinity if and only if the limit as n goes to infinitiy of bn is positive infinity.

Here is what I have for proving the first way:

Suppose that limit of {an/bn} = L > 0, and that lim as n approaches infinity of an is positive infinity. Let e > 0. Then there exists N (a natural number) such that for all n>N |(an/bn) - L| < e. The |an/bn| - |L| <= |an/bn - L| < e. Thus, an/bn - L < e since {an}, {bn}, and L > 0. Then an/bn < e+L and an < bn(e+L). Since lim of an is infinity, then for any M > 0, there exists N1 (a natural number) such that an > M for all n > N1. Thus M < an < bn (e+L) for all n > max (N, N1), and bn(e+L) > M. Then bn > M/(e+L) for all n > max(N,N1). Thus, bn diverges to positive infinity.

Please let me know if this proof is acceptable, and how to prove the other way (If bn goes to infinity, then an goes to infinity). Thank you in advance for any assistance.

Solution Preview

Let "oo" denote infinity.
Your proof is only the "if" part. That is: if a_n->oo as n->oo, then b_n->oo as n->oo.
Here is the proof of "only ...

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