Guidance with a floor function proof

Question

I need some guidance in helping me solve this proof. Here is the question:

Give a proof by cases that $\lfloor 4x \rfloor = \lfloor x \rfloor + \lfloor x + 1/4 \rfloor + \lfloor x + 1/2 \rfloor + \lfloor x + 3/4 \rfloor$

Answer 1

HINT: Let $\alpha=x-\lfloor x\rfloor$, the fractional part of $x$. Your four cases are $0\le\alpha<\frac14$, $\frac14\le\alpha<\frac12$, $\frac12\le\alpha<\frac34$, and $\frac34\le\alpha<1$. Investigate $\lfloor 4x\rfloor$ separately for each case; it may be helpful to write $x$ as $\lfloor x\rfloor+\alpha$.