Which theorem is it, or can somebody help with a proof to it? $$\lim_{n\to \infty} a_n=0  \iff \lim_{n\to \infty} \frac{a_{n+1}}{a_n}<1 \text{ when } a_n>0 \text{ and } \frac{a_{n+1}}{a_n}>0.$$

How about this one:
$$
a_n = \frac 1 n.
$$
Then we have
$$
\lim_{n\to\infty} a_n = 0 \text{ and } \lim_{n\to\infty} \frac{a_{n+1}}{a_n} = 1.
$$
So the proposed theorem is false.
