Answer to Question #155668 in Calculus for Vishal

Question

If x is a real number, prove that there are integers p, q ∈ Z such that p < x < q.

Answer 1

Let us assume that there is no $q\in \Z$ such that $x<q$ . This implies that $x\geqslant q$ .

$\Rightarrow$ This means that x is an upper bound for $\Z$ .

But, this contradicts the fact that $\Z$ is unbounded above.

$\therefore x<q$

Similarly,

Let there is no $p\in\Z$ such that $x>p$ . This implies that $x\leqslant p$ .

$\Rightarrow$ This means that x is a lower bound for $\Z$ .

But, this contradicts the fact that $\Z$ is unbounded below.

$\therefore x>p$

So combining the previous two results we have,

$\forall x\in\R$ , $\exists$ $p,q\in\Z$ such that,

$$p<x<q$$