# Where does this inequality come from?

1. Sep 18, 2007

### michaelxavier

In Rudin's Principles of Mathematical Analysis, 3rd ed., I encountered the following on p. 10, and I'm not really sure where it comes from. I'll write it just as it is shown in the book.

The identity
b^n-a^n=(b-a)(b^(n-1)*a^0 + b^(n-2)*a^1 + ... + b^1*a^(n-2) + b^0*a^(n-1))
yields the inequality
b^n-a^n<(b-a)n*b^(n-1)
when 0<a<b

I understand where the indentity comes from. I'm just confused about the inequality, and I was curious. Thanks for sharing any ideas.

2. Sep 18, 2007

### morphism

0<a<b implies, for instance, that b^(n-2)a < b^(n-1). Do you see it now?

3. Sep 18, 2007

### michaelxavier

Oh, of course. Thanks so much!