The sequence $\left\{ \frac{1}{n}\right\}$ is bounded since $0\lt \frac{1}{n}\leq 1$ and $1$ is its lest upper bound (lub) of the sequence $\left\{ \frac{1}{n}\right\}$.
According to definition of lub, a real number $M$ is called the lub of a real sequence $\left\{ f(n)\right\}$ if the following conditions hold:
(i) $f(n)\leq M$ for all $n\in \mathbb{N}$.
(ii) for each $\epsilon>0$, there exists a real natural number $k$ such that $M-\epsilon<f(k)$
For the sequence $\left\{ \frac{1}{n}\right\}$, $M=1$, if we choose $\epsilon=\frac{1}{4}$, then there does not exist any natural number $k$ such that $ \frac{1}{k}>M-\epsilon=1-\frac{1}{4}=\frac{3}{4}$.
Can anyone clarify this confusion?