Let {an} and {bn} be sequences and suppose that an <= bn for all n and that an->

ID: 2980200 • Letter: L

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Let {an} and {bn} be sequences and suppose that an <= bn for all n and that an-> infinity. Prove that bn-> infinity

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FOLLOW THIS By hypothesis, given any e > 0, there exists N > 0 such that |(an/bn) - L| = N. ==> an/bn bn > an/(2L) for all n > N. Also, an/bn > L - e > L ==> an > L * bn for all n > N. =========================== Suppose that lim an = infinity. Then for every M > 0, there exists N' > 0 such that an > M for all n >= N'. Choosing N* = max{N, N'} we have that bn > an / (2L) > M/(2L) for all n > N*. Since M is arbitrary, this shows that lim bn = infinity. ==================== Now, suppose that lim bn = infinity. Then for every M > 0, there exists N' > 0 such that bn > M for all n >= N'. Choosing N* = max{N, N'} we have that an > L * bn > ML for all n > N* Since M is arbitrary, this shows that lim an = infinity

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Let {an} and {bn} be sequences and suppose that an <= bn for all n and that an->

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