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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 $\varepsilon=1$).2012-10-10

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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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    @N.S.: No need: by hypothesis these are positive sequences.2012-10-10