Could someone help me on this Proof please?

Question

We went over this question today in class and I don't understand it. Could someone help?

Let a and b be integers. Prove that a<b if an only if a≤b-1

What I did was rename a and b, x and x+1 respectively. I then plugged those into a≤b-1 where, after doing the math, x=x. I am wondering if I am on the right track? Thanks

Answer 1

Pippi,

This is an if and only if statement, so you need to prove both directions. First assume a and b are integers and a < b. Since a and b are integers and a < b, the largest a can be is b - 1, so a a ≤ b - 1.

Conversely, suppose a and b are integers and a ≤ b - 1. Then a ≤ b - 1 < b, so a < b. That is all.

Andrew