Answer to Question #155590 in Calculus for shiam

Question

Exercise 1. If x is a real number, prove that there are integers p, q ∈ Z such that p < x < q.

Answer 1

Let q=p+1, then

$$p<x\le q$$

$$p<x\le p+1$$

If x=p+1, then we will take

$$q=p+i\times 1$$

where i is any natural nomber.

Then we will have any I for will be true next:

$$p<x<p+i\times 1$$

For example,