Given $\lim _{n\to \infty} a_n = L$ ($L>0$), how to prove that $\lim_{n \to \infty} 1/a_n = 1/L$ without knowing the expression of $a_n$?
How to formally prove that if $\lim \limits_{n \to \infty}a_n=\infty$, then $\lim \limits_{n \to \infty}\frac{1}{a_n}=0$.