Question 1074715
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Counterexample for (a) 

&#8970;1.6*1.7&#8971; ?&#8804;? &#8970;1.6&#8971;*&#8970;1.7&#8971; 
&#8970;2.72&#8971; ?&#8804;? &#8970;1&#8971;*&#8970;1&#8971; 
2 ?&#8804;? 1*1
2 ?&#8804;? 1

False, so (a) is disproved.

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Proof for (b) 

Let i = the integer part of x
Let f = the fraction part of x

Let j = the integer part of y
Let g = the fraction part of y

Then x = i+f, 0 < f < 1
     y = j+g, 0 < g < 1


&#8968;x*y&#8969; ?&#8804;? &#8968;x&#8969;*&#8968;y&#8969;
&#8968;(i+f)*(j+g)&#8969; ?&#8804;? &#8968;i+f&#8969;*&#8968;j+g&#8969;

&#8968;ij+ig+fj+fg&#8969; ?&#8804;? (i+1)*(j+1)

&#8968;ij+ig+fj+fg&#8969; ?&#8804;? ij+i+j+1

since ig < i and fj < j and fg < 1, that's true.

So the theorem is proved for positive real numbers. 

[it's not true for negative real numbers though.]

Edwin</pre>