About floor function

Question

I am stuck with these two floor function problems.

Please help me

1. Let r be a real number, and n be a positive integer.
Prove $[r]+[r+\frac1n]+...+[r+\frac{n-1}n]=[nr]$

2. Let S be set of integers given by $[\alpha x]$ and $[\beta x]$ for x=1,2,3,...
Prove that S consists of every integer, each appearing exactly once, iff $\alpha$ and $\beta$ are positive irrational numbers such that $\frac1\alpha+\frac1\beta=1$

Thank you

Answer 1

Part 1 hint:

Consider the value of $r-[r]$.