Theorem 5.3. Let x and y be real numbers.Then lxyl lxllyl. Proof. 问题3：这种证明方法 First,suppose that x 0 and y 0.Then xy 0 and we have 为什么被称为分情形证 Ixyl =xy,xl=x,and lyl =y.Therefore, 明法？ Ixyl =xy Ixllyl, and we have established the result in this case. Second,suppose that x<0 andy 0.Then xy 0 and we have Ixyl xy,Ixl =-x,and lyl =-y.Therefore, 问题5：有的时候，我 =xy=(-x)(-0=xl, 们在证明时会用到“不 and we have the result for this case as well. 失一般性”这个词，你 Third,suppose that either x=0 ory=0.Then xy =0 and we 理解这是什么意思吗？ have lxyl =0,and either x=0 or lyl =0.Therefore, =0=xl, establishing the result in this case too. For our final case,suppose that one number is positive and the other is negative.Thus,we may assume that x<0 and y >0.Then 问题4：这种证明方法 xy 0 and we have lxyl =-(xy),lxl =-x,and lyl y.Therefore, 最“令人担心”的是什 x=-(0=(-x)y=xl 么？ We have now established the result for all four possible cases and we may conclude that lxyl =Ixllyl for all real numbers x and y.问题3：这种证明方法 为什么被称为分情形证 明法？ 问题4：这种证明方法 最“令人担心”的是什 么？ 问题5：有的时候，我 们在证明时会用到“不 失一般性”这个词，你 理解这是什么意思吗？