Elementary Number Theory and Its Application, 6th Edition by Kenneth H. Rosen Exercises 1.1.13

13. Show that [x + y] >= [x] + [y] for all real numbers x and y.

Solution: We consider the following inequation

[x+y]>=[x]+[y]
We know by the definition of greatest integer function that
[x]<=x ……(1)
[y]<=y ……(1)
Adding the inequations in (1) gives us
[x]+[y]<=x+y
Since greatest integer function preserves the order, therefore,
[x+y]>=[[x]+[y]]
[x+y]>=[x]+[y]
Hence, we have proved that
[x+y]>=[x]+[y]

 

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