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Question:

Show that floor(5*x) <= 4 + 5*floor(x)?

Show that floor(5*x) <= 4 + 5*floor(x)?

Answer:

floor(x) is x rounded down to the next integer. So floor(8.3) = 8, floor(-4.6) = -5, floor(11) = 11. Let x = d + r, where d is an integer and r is a decimal with 0 <= r < 1. floor(5*x) = floor(5d + 5r) = 5d + floor(5r) <= 5d + 4 = 4 + 5*floor(d) <= 4 + 5*floor(x). EDIT I had a typo above when defining d and r, it is now correct.

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