Supremum of sets containing inequalities. Let $r \in \mathbb{R}$ be fixed. Prove that:
a) $\sup\{q \in \mathbb{Q}: q \leq r\} = r$.
b) $\inf\{q \in \mathbb{Q}: q \leq r\} = r$.
It looks so strange for me that the supremum equals the infimum, Could anyone help me or at least say that the problem contains a mistake? 

I think you should have for (b): $\inf \{q \in \mathbb{Q}: q \geq r \} = r$ as $\inf \{q \in \mathbb{Q}: q \leq r \} = -\infty$ 
You might even say that the latter expression is undefined, regardless there is certainly no real $r = -\infty$.
