I have the following to prove:
$$\lfloor 3x\rfloor = \lfloor x\rfloor + \left\lfloor x+\frac 13 \right\rfloor + \left\lfloor x+\frac 23 \right\rfloor $$ The definition of a floor function is: $ \lfloor x \rfloor = n \le x \lt n+1 $
So my first instinct was to do $ \lfloor 3x\rfloor=3 \lfloor x\rfloor $ and then let $n=\lfloor x\rfloor$ so basically we get $3n$. But if I were to replace both sides of the equation with $n=\lfloor x\rfloor$ I get : $$3n = n + \left\lfloor x + \frac 13 \right\rfloor + \left\lfloor x+\frac 23 \right\rfloor =3n+\frac 33=3n+1$$
But I don't know what to do with this and I'm not sure if this is a formal way of doing these types of proofs.