Hello!

I want to prove the following:

where is the distance to the nearest integer. and k is an integer.

Here is what I did so far:

Since I know there exists a c with . Thus I can conclude that where

Now I got

I know because of that

I should now look at the cases that k is even and odd and greater or less than zero. But I donīt get to anything when I look at the last inequality to calculate The case k=0 is trivial.

Plan is to look at k greater and less 0 and k even and odd for this case and then also do the same thing for the right hand side and compare those values and it should be the same result. Then I am done. Can anyone give me a hint on how to progress from here?

Thanks.