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

The forward implication is wrong; take for example $a_n=\frac 1 n, \lim_{n\to \infty} \frac {a_{n+1}} {a_n}=1$. The other implication however is true (if the limit of $\frac {a_{n+1}} {a_n}$ exists) and I used to call it "ratio criterion"; the proof is not difficult. First, note that $a_{n+1}<a_n$, so the sequence (which is also positive $\forall n\in \mathbb N$) has a limit. Moreover, you can say that$\frac {a_{n+1}} {a_n} < M $ for n big enough, where M is a constant $<1$. So $a_{n_0+k+1}< M a_{n_0+k} < M^2 a_{n_0+k-1}<..., a_1$ you can write $0< a_{n_0+k+1} < M^k a_{n_0} $, so $0< \lim_{k\to \infty} a_{n_0+k+1} < \lim_{k\to \infty} M^k a_{n_0}$, but we have M<1, so $a_n\to0$.
