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