Let l be an element of R, c be an element of R, c>0. Show that the following two statements are equivalent:
a)for every epsilon>0 there exists N that's an element of the natural numbers for every n>or=N such that |xn-l|< epsilon
b)for every epsilon>0 there exists N that's an element of the natural numbers for every n>or=N such that |xn-l|< (c)epsilon

I came across this problem in my textbook while studying, but can't figure out how to approach it. Any help would be great!

2. We may use the fact that if $c,\epsilon>0$, then both $c\epsilon>0$ and $\frac{\epsilon}{c}>0$.

3. What if c is less than 1?

4. That would be okay; the conclusions of

\begin{aligned}
c,\epsilon>0&\rightarrow c\epsilon>0\\
c,\epsilon>0&\rightarrow \frac{\epsilon}{c}>0
\end{aligned}

would still follow.

5. i've now tried the proof several times and still can't show that both statements imply each other, can you provide some more guidance?

6. To prove (b) from (a), we may apply theorem (a) to the positive number $\epsilon'=c\epsilon.$