Ok, this might be a silly/easy question but I have to make sure. Let say we have a seq of pos. numbers $\displaystyle \{x_n\}$ such that $\displaystyle x_n<x_{n+1}$ $\displaystyle \forall n$ and $\displaystyle lim_{n\rightarrow \infty}(x_n/x_{n+1})=1$. Can I then, knowing the above, conclude that $\displaystyle \sup\{x_n/x_{n+1}; n\in\mathbb{N}\}=1$?