Consider set $D$:
$$D = \left\{m + \frac1n\,\middle|\,m,n\in\mathbb{N}\right\}$$
So, the supremum of this set does not exist because this set is NOT bounded above. I'm confused about the infimum in this case. The set is bounded below but I'm confused about what the greatest lower bound is.
(I am a new user so if you have any suggestions about my question writing, feel free to drop a comment)
Hint: Every element of $D$ is greater than $1$ but, if $a>1$ there is always an element $d\in D$ such that $d<a$.