Floor Functions Proof

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I have a floor function proof that I need to do. The arithmetic of the actual problem isn't my problem. It is the way to do the direct proof. I don't know how to prove this. Prove that if x is a real number, then ⌊2x⌋ = ⌊x⌋ + ⌊x + 1/2⌋. I already know for any x this is true but how do you do the proof?