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?

Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? — 5xum, Mar 10 '16 at 09:20

score 2 · Accepted Answer · answered Mar 10 '16 at 09:21

2

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 $\lfloor x+k\rfloor$.

answered Mar 10 '16 at 09:21

Hagen von Eitzen

374,180

score -1 · Answer 2 · answered Mar 10 '16 at 09:38

-1

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

answered Mar 10 '16 at 09:38

Torn

235

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

2 Answers2