Suppose we have a sequence $\{\frac{b_n}{n}\}_{n\in \mathbb{N}}$ and we know that $$ 0 \leq \frac{b_n}{n}\leq 1 \text{ }\forall n \in \mathbb{N} $$ What can we conclude about $\lim_{n\rightarrow \infty} \frac{b_n}{n}$?
My impression is that we can just say the following
"If $\exists \lim_{n\rightarrow \infty} \frac{b_n}{n}$, then it should be equal to $L\in [0,1]$"