My first year Analysis textbook at university includes examples to grasp the concepts of infimum, supremum, maximum, minimum, lower bound and upper bound in set theory for subsets of the real numbers $\mathbb{R}$. One of such examples is the set:
$$S=\{x\in \mathbb{Q}\;|\;x^2<2\}$$
for which the textbook says that "there does not exist a maximum, the upper bounds are all real numbers greater than or equal to $\sqrt{2}$, and the supremum is $\sqrt2$".
Now, it also says that for this same subset, "there is no minimum, there are no lower bounds and thus there is no infimum": to me, it would seem like there is no minimum (since one could approach $-\sqrt2$ infinitely by smaller and smaller rational numbers part of the set), with the infimum being $-\sqrt2$ and lower bounds all real numbers smaller than or equal to $-\sqrt2$. Am I wrong, or is the textbook wrong?
Best Answer
Looking at $S$ as a subset of $\mathbb Q$ it has lower bounds, but has no infimum (i.e. greatest lower bound).
Looking at $S$ as a subset of $\mathbb R$ it does have an infimum which is $-\sqrt2\in\mathbb R$.