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**captainjapan** Recall that the floor of a real number x, i.e. [x], is defined to be the largest integer d such that d <= x

Proposition. For any real numbers x and y, [x + y] >= [x] + [y]

(a) Explain briefly why it's not true if you replace >= with =

(b) Prove that this claim is true.

Notice that the definition of floor folds together two properties. If t is the floor of z, then (i) t <= z and (ii) for any integer d, if d <= z, then t >= d.

It may help to give short names (e.g. like d) to expressions like [x + y].