Proving the implication $\lim_{n\to\infty} a_n a_{n+1} =0 \Rightarrow \lim_{n\to\infty} a_n=0$ Let $\{a_n\}$ be a sequence such that for all $n$, $a_n>0$. I have to prove that if $\lim_{n\to\infty} a_n a_{n+1} =0$ then $\lim_{n\to\infty} a_n=0$. Here's my reasoning: $\forall\varepsilon>0,\exists k$ such that $\forall n>k$ we have $|a_n a_{n+1}|=a_n a_{n+1} < \varepsilon$ and thus $a_n=|a_n|=|a_n-0| < \frac{\varepsilon}{a_{n+1}} \Rightarrow \lim_{n\to\infty} a_n=0$. However I don't feel comfortable with the $a_{n+1}$ on the r.h.s. What do you think?

As said that's not correct. I think you should better argue by contradiction that $\lim_{n\to\infty}a_n\neq 0$. WLOG the limit exists and $$\lim_{n\to\infty} a_n=\lim_{n\to\infty}a_{n+1}=l>0\Rightarrow \lim a_na_{n+1}=l^2>0$$ in contradiction which means that necessarily $\lim_{n\to\infty}a_n=0$ . 
