Metric Spaces: Continuous, Unbounded Functions The following question is from Fred H. Croom's book "Principles of Topology"


Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous, unbounded function. Show that there is a number $t_0$ for which $\{f(nt_0):n$ an integer $\}$ is an unbounded set. 


For reference: $D(A)=\sup\{d(x,y):x,y\in A\}$. Alternatively, $D(A) = \sup A-\inf A$
The idea I have in mind is as follows. We can have $A$ = $\{f(nt_0):n$ an integer $\} \subset \mathbb{R}$. We would have to show that the $D(A)$ does not exist. To do this, I would have to show that $\sup A$ and $\inf A$ do not exist. Am I on the right track? How does continuity apply in this case?

I want to thank you for taking the time to read this question. I greatly appreciate any assistance you provide.  



The idea I have in mind is as follows. We can have $A$ = $\{f(nt_0):n$ an integer $\} \subset \mathbb{R}$. We would have to show that the $D(A)$ does not exist.


The problem with that proposed solution is that it can work only if you can show that for every number $t_0$, $D(A)$ does not exist.  The problem is to show that for some $t_0$, a conclusion holds.  To say that it holds for every $t_0$ is a substantially stronger claim.
