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## Homework Statement

Prove that for any real number x, if x - floor(x) < 1/2, then floor(2x) = 2 floor(x)

## Homework Equations

## The Attempt at a Solution

Assuming that x is a real number. Suppose that x - floor(x) < 1/2

Multiplying both sides by 2, 2x < 2 floor(x) + 1

from the definition, 2 floor(x) <= 2x

2 floor(x) is an integer, and 2 floor(x) <= 2x < 2 floor(x) + 1

Implying floor(2x) = 2 floor(x)

My question is, how does the proof imply the result? I don't see how the logic works. Thank you for any help you can provide in advance.