Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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, $\lim y_n = \infty$.

share|improve this question
1  
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

1 Answer 1

up vote 0 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 $\dfrac{x_n}{y_n}\ge\dfrac{x_n}M$.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.