Understanding Rudin Theorem 1.20 (b): Integers and Real Numbers Explained

[Dschumanji]

I understand the proof except for the following:

Suppose that -m₂ < nx < m₁ for positive integers m₁, m₂, n, and real number x.

Then there is an integer m with -m₂ ≤ m ≤ m₁ such that m-1 ≤ nx < m.

It definitely sounds reasonable, but it seems like a big jump in logic.

 

[jgens]

Let m be the least integer that is strictly greater than nx. It is a triviality to verify that this integer has the desired properties.

 

[mabello]

Simple proof

Hi there,
I have attached a simple demonstration of the bit you are asking.
Let me know if it is clear now.
I hope it helps