 
Summary: Sample Problems 2, Math 8 Summer Session C, 2010
Here are some problems like the ones that will appear on midterm 2. You should also be
able to do all of the exercises in chapters 4 and 8.
You are welcome to use results proved in the text. If you do so, then you must briefly and
precisely state the result you are using.
1 Suppose u > v > 1. Using Rules 4.1 to carefully justify your claims, prove that u2 > v.
Proof Since u > v and v > 1, 4.1(5) yields u > 1. Since v > 1 and 1 > 0, we have v > 0
(also by 4.1(5)). So we may apply example 4.3, which gives uv > v. Since u > v and
v > 0, it follows that u > 0 and so, by 4.3, u2 > uv. Finally, by transitivity, we have
u2 > v.
2 Suppose that x, y R. Using Rules 4.1 to carefully justify your claims, prove that
(x2
+ y2
)/2 xy
.
Proof We proved in the text that if x R, then x2 0. It follows that (x  y)2 0. But
(x  y)2 = x2  2xy + y2, and so 4.1(3) implies that x2 + y2 2xy. If the inequality is
strict, then we may apply example 4.3 to get (x2 + y2)/2 > xy. If, on the other hand, we
have x2 + y2 = 2xy, then we simply multiply by 1/2 to get (x2 + y2)/2 = xy. Therefore
we have shown
