There are several questions related to this problem on this cite. And there are good answers. But I did not find what I search.
DEFINITION: Let E be a subset of $\mathbb{R}$. We say that $M\in \mathbb{R}$ is a least upper bound for E iff (a) M is an upper bound for E and (b) any other upper bound for E must be larger than or equal to M.
LEMMA: A number $u\in \mathbb{R}$ is the supremum of a non-empty subset S iff it has the following properties:
- there are no elements $s\in S$ with $u<s$
- if $v<u$, then there is an element $s_v\in S$ such that $v<s_v.$
PROBLEM 1.1: Show that $$\sup\{x\in \mathbb{Q}:x>0, x^2<2\}=\sqrt{2}.$$
In the solution to this problem is written: Set $s$ to be the supremum. We will now show that for any positive integer n $$(s-1/n)^2\leq 2\leq (s+1/n)^2.$$
QUESTION: What idea does stay behind this double inequality? The right inequality is (b) condition from the definition. But what is the left inequality? I would like to say that it is the second property from the lemma. But there is the strict inequality.
EDIT: 