Proving that $\lfloor x + k \rfloor=\lfloor x\rfloor + k$

Question

How to prove that $\lfloor x + k \rfloor=\lfloor x\rfloor + k$ when $k$ is an integer?

Answer 1

We have $\lfloor x\rfloor \le x<\lfloor x\rfloor +1$, hence $\lfloor x\rfloor +k\le x+k<\lfloor x\rfloor+k +1$, so that the integer $\lfloor x\rfloor +k$ fulfills the defining property of $\lfloor x+k\rfloor$.

Answer 2

1) ⌊x⌋ = x - {x}
2) {k} = 0
3) ⌊x + k⌋ = x + k - {x + k}
According to point 2:
{x + k} = {x}
And from there on just replace the parts in point 3.
x + k - {x + k} = x + k - {x} = x - {x} + k = ⌊x⌋ + k