Asymptotics of expression involving floor functions

Question

Consider two functions $f_1(n)$ and $f_2(n)$ that grow to infinity at the same speed, say $$\lim_{n\to \infty} \frac{f_1(n)}{f_2(n)}=c $$ for some $c>0$. I am studying the squared difference of $f_1$ and $f_2$. Can the squared difference of the floor functions of $f_1$ and $f_2$ be estimated by that of the original functions? In other words, my question is: does there exist some constant $C>0$ such that $$ ( \lfloor f_1(n) \rfloor - \lfloor f_2(n) \rfloor )^2 \leq C (f_1(n) - f_2(n) )^2 $$ for n large enough? It is obvious that $\lfloor f(n) \rfloor$ grows like $f(n)$, but the difference term makes things more complicated.

Answer 1

This holds if $c\ne1$. If $c=1$, then you need to ensure that the difference goes to infinity. Otherwise, you could have a situation like $f_1(n)=n$, $f_2(n)=n-1/n$.