Prove this number fact Prove that $x \neq 0,y \neq 0 \Rightarrow xy \neq 0$. 
Suppose $xy = 0$. Then $\frac{xy}{xy} = 1$. Can we say that $\frac{xy}{xy} = 0$ and hence $1 = 0$ which is a contradiction? I thought $\frac{0}{0}$ was undefined. 

Sorry the answer is a bit late, but I just saw this post and happened to have Rudin's book on me so I've taken a look at the proposition the OP was referring to. 
1.16 (b) If $x\neq 0$ and $y\neq 0$ then $xy \neq 0$. 
the proof essentially goes as follows 
Suppose $x\neq 0$ and $y\neq 0$, and suppose for contradiction that $xy = 0$. Since $x, y\neq 0$ we know that $\frac{1}{x}, \frac{1}{y}\in F$, Hence $1 = (\frac{1}{x})(\frac{1}{y})xy = (\frac{1}{x})(\frac{1}{y})0 = 0$ a contradiction. 
The fallacy given in the OP is that since we assume that $xy = 0$ we cannot write $\frac{xy}{xy}$ because we are clearly dividing by zero. However, we can write $(\frac{1}{x})(\frac{1}{y})$ because we know both are in the field (and hence their product is as well) since $x, y \neq 0$. 
I hope this helps even though it's late 
