Show that if $\lfloor x+a \rfloor$ = $\lfloor x+b \rfloor, \forall x \in \Bbb R$ then $a=b$; is showing that $x+a=x+b$ enough?

Question

If i show that $x+a=x+b$ only if $a=b$, does that prove that the above is also true?

$ x+a=x+b \iff x+a-x-b=0 \iff a-b=0 \implies b=a$ also is this any good?

Answer 1

That argument isn't going to work, since at no point are you using any properties of the floor function.

What will work is something like this (only a hint):

If $a \ne b$ then we can assume (without loss of generality) that $a < b$, and then we can take some $y$ such that $ a < y < b$. Can you then find some expression for $x$ which will make sure that $x+a$ and $x+b$ will have different integer parts?