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?

if limit exist then $\lim_{n\to\infty}a_{n}=\lim_{n\to\infty}a_{n+1}$ thus $\lim_{n\to\infty}a_na_{n+1}=\lim_{n\to\infty}a_{n}^2=0$ thus $\lim_{n\to\infty}a_n=0$ if limit does not exist we cannot comment on $\lim_{n\to\infty}a_n$
