I'm trying to prove that if $\;\frac{\sin{a_n}}{a_n}\rightarrow 1$ then $a_n\rightarrow0$, assuming $a_n\neq 0$ for all $n$.
I think this is easy enough to show as follows: first, prove $f(x)=\frac{\sin(x)}{x},f(0)=1$ has a global maximum at 0, then assume by negation that not $a_n\rightarrow0$, and reach a contradiction with epsilon-delta gymnastics. But this will turn out to be a long and somewhat messy proof.
Is there a more elegant way to prove this?
Addendum: this isn't a homework question (I'm not sure if it looks like one), so if possible, please give the full details of your answer.
Thanks!