Q:

Cauchy Sequences--is the floor function of a Cauchy sequence also a Cauchy sequence?

Okay so say you have some Cauchy sequence (a_n). And c_n=[[a_n]], where [[x]] refers to the greatest integer less than or equal to x. 
Is c_n also a Cauchy sequence? 
This is what I've got so far, not much cause this is really stumping me:
+I know that a_n converges to some real number, say a. 
+I was considering a_n=1-(1/n). That gives c_n={0,0,0,0,....}. Is a sequence of all one constant integer a Cauchy sequence?
+If I can't prove that c_n is a Cauchy sequence I need to show a counterexample. 
+I can't figure out how to use the Cauchy sequence definition in this problem, not sure if that's the right way to go about it or not. 
Thank you for your time!

A:

HINT: Consider the sequence $\langle x_n:n\in\Bbb N\rangle$ defined by 
$$x_n=\begin{cases}
0,&\text{if }n\text{ is even}\\
-\frac1n,&\text{if }n\text{ is odd}\;.
\end{cases}$$

