Consider the sequence $\{ y_{n} \}$ of real numbers such that $\sup\{\left | y_{n} \right |:n\in \mathbb{N}\}=4$. Find $\inf \left\{\frac{\left | y_{n} \right |}{n}:n\in \mathbb{N} \right\}$
Since $4=\sup\{\left | y_{n} \right |:n\in \mathbb{N}\}$, then it follows that $0\leq \frac{\left | y_{n} \right |}{n}\leq \frac{4}{n}$, and hence $$\lim_{n \to \infty }\ \frac{\left | y_{n} \right |}{n}=0.$$ I feel like the infimum of the set: $\{\frac{\left | y_{n} \right |}{n}:n\in \mathbb{N}\}$ should be zero, but a rigorous proof eludes me. It's obvious that $0$ is a lower bound for the set, but I don't know how to prove that it is the greater lower bound for the set. Any help is appreciated!