Due Tuesday (Aug 31) 6pm. 2 points each.

1. Someone claims that he has found a smallest positive rational number, but would not tell you which number it is, can you prove that this is impossible? (Optional extra question: can you prove that there is no smallest rational number among all rational numbers that are larger than $\sqrt{2}$?)

2. Prove that, if $r$ is a non-zero rational number, $x$ is an irrational number, then $r + x$ and $rx$ are irrational. (UPDATE: if $r=0$, of course $rx=0$)

3. Prove that there is no rational number whose square is $20$.

4. Read Ross Section 1.2 about mathematical induction, and prove that $7^n - 6n - 1$ is divisible by $36$ for all $n \geq 1$.