- #1
rudders93
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Homework Statement
Without using logarithms, prove that [itex]a^{n}\rightarrow0[/itex] as [itex]n\rightarrow\infty[/itex] for [itex]|a|<1[/itex] by using the properties of the sequence [itex]u_{n}=|a|^{n}[/itex] and its subsequence [itex]u_{2n}[/itex].
The Attempt at a Solution
I'm really stuck on this. One thing I thought of doing was to prove that the sequence is strictly decreasing as [itex]u_{n+1}-u_{n}=|a|^{n+1}-|a|^{n}=|a|^{n}(|a|^{n}-1)<0[/itex] as [itex]|a|<1[/itex]. Also we know it's bounded within interval (0,1) and so by the Monotonic Sequence theorem it must converge to some limit. Then using standard limits we can prove that it converges to 0, but I think that's kind of cheating the proofHowever, not sure how to do this using the properties of subsequences. How can I do it using that?
Thanks!