Find the infimum,supremum,maximum and minimum of the following set or prove non existence: $$E = \{\frac{p}{q} \in \mathbb{Q}: p^{2} < 2q^{2}, p,q >0 \}.$$
This question is solved in the following:
But I did not understand why he said that $\omega$ (as he defined in the second picture) is a rational number, Could anyone explain this for me?
On
Another way to approach, e.g., the question of what the maximum is in this set: You are searching for rational numbers $p/q$ with both $p, q > 0$ for which $p^2 < 2q^2$. The last inequality is equivalent to $(p/q)^2 < 2$, which, in turn, is equivalent to $p/q < \sqrt{2}$.
So, you are trying to find the maximum rational $p/q$ that is less than $\sqrt{2}$; but clearly this does not exist, for you can take a rational sequence that is monotonically increasing and converges to $\sqrt{2} \not\in \mathbb{Q}$, which means that for any rational less than $\sqrt{2}$ you can go far out enough in this sequence to find an even greater rational that is still less than $\sqrt{2}$.
A concrete example of such a sequence can be found by looking at the decimal expansion of $\sqrt{2}$:
$$\sqrt{2} = 1.414213\ldots$$
Next, use rationals of the form "natural number" over "power of ten" corresponding to the above:
$$1/1, 14/10, 141/100, 1414/1000, 14142/10000, 141421/100000, 1414213/1000000, \ldots$$
Using this monotonically increasing sequence that converges to $\sqrt{2}$, we find that the maximum of the given set does not exist.
I think that $[{}\cdot {}]$, as it appears in the first numerator in the definition of $w$, is the floor function (also called the greatest integer function). That makes $[(n+1)s]$ an integer (the greatest integer $m$ such that $m \leq [(n+1)s]$, by definition), so all numerators and denominators in the definition of $w$ are integers, which makes $w$ rational.