I'm trying to prove the ratio test for sequences. Here's what I got:
If $ \lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = L < 1 $ and $ a_n>0 \;\ \forall n $ then $ a_n $ is bounded below by $0$. Also there's $N$ so that forall $n>N$, $a_{n+1}<a_n $. Therefore, the sequence is decreasing and bounded below so it must converge.
Now, according to the test, $ \lim \limits_{n \to \infty} a_n = 0 $. Why is that?