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Let $(x_n)$ and $(y_n)$ be sequences of positive numbers such that $\lim \frac{x_n}{y_n} = 0$.

a) Show that if $\lim x_n = \infty$, then $\lim y_n = \infty$.

b) Show that if $y_n$ is bounded, then $\lim x_n = 0$.

For a, could I say, there exists a $N$ such that $\frac{x_n}{y_n} < 1$ for $n > N$. So $y_n > x_n$ for any $n > N$?

So, $\lim y_n = \infty$.

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Yes, that works. Part (b) can be done in a similar way (except that you can't just choose $\varepsilon=1$). – wj32 Oct 10 '12 at 3:33
up vote 1 down vote accepted

Your argument for (a) is fine. For (b), let $M$ be such that $y_n\le M$ for all $n$, and note that $\dfrac{x_n}{y_n}\ge\dfrac{x_n}M$.

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For $(b)$ I would add absolute values though, otherwise there might be issues if some, but not all, of $x_n, y_n$ and $M$ are negative... – N. S. Oct 10 '12 at 4:08
@N.S.: No need: by hypothesis these are positive sequences. – Brian M. Scott Oct 10 '12 at 4:20

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