# Rudin 1.21

1. Sep 4, 2008

### Unassuming

In Rudin 1.21 he says the following in the midst of proving a theorem,

"The identity b$$^{n}$$ - a$$^{n}$$= (b-a)(b$$^{n-1}$$ + b$$^{n-2}$$a + ... + a$$^{n-1}$$) yields the inequality

b$$^{n}$$ - a$$^{n}$$ < (b-a)nb$$^{n-1}$$ when 0 < a < b"

I can understand that it is less than, but I cannot understand how it is coming (yielding) from the identity.

Any explanation would be greatly appreciated.

2. Sep 4, 2008

### Vid

Try seeing what happens to the second term on the right side when b=a.

3. Sep 5, 2008

### sutupidmath

Vid's point is that:

$$b^n-a^n=(b-a)(b^{n-1}+b^{n-2}a+.....+ba^{n-2}+a^{n-1})<(b-a)\underbrace{(b^{n-1}+b^{n-2}b+.....+bb^{n-2}+b^{n-1})}=(b-a)nb^{n-1}$$ since a<b

4. Sep 5, 2008

### Unassuming

...that's clever. Thank you.