Are all proper subsets of $\mathbb R$ bounded?
Solution 1:
Not every proper subset of $\mathbb{R}$ is bounded. For instance all the positive reals is not a bounded set, or for another example, $\mathbb{R}\setminus \{10\}$ is also unbounded. You've likely defined $\inf(A)$ for sets that are not bounded from below to be $-\infty$, but check this in your lecture notes.