**15. Show that if x and y are positive real numbers,then [xy] >= [x] [y]. What is the situation when both x and y are negative? When one of x and y is negative and the other positive?
**Solution: We have to prove the inequality

[xy]>=[x][y]

Where x and y are positive real numbers.

We prove the above inequality by making assumptions.

Let us assume that

x=a+r

y=b+s

Where a,b are integers and r,s are real numbers such that 0<=r,s<1

Now, using the values of x and y , we get,

[xy]=[(a+r)(b+s)]

[xy]=[ab+as+br+rs]

[xy]=ab+[as+br+rs]

Whereas

[x][y]=[a][b]

[x][y]=ab

This is because a,b are integers and

[r]=0

[s]=0

Where 0<=r,s<1

Thus we have

[xy]<=[x][y]

If one of x and y is positive and the other negative, then depending on what the numbers are, the order sign can go in either direction.