Continuous functions and converging sequences If $f$ is continuous on $[a,b]$ and $\{ x_n\}$ is a sequence in $(a,b)$, then $\{f(x_n)\}$ has a convergent subsequence. 
Is it true or false?


I think this is false but I am not sure of an example that would show it...



Hints:


*

*$[a,b]$ is compact.

*$f$ is continuous, so $f([a,b])$ is compact.

*$\{f(x_n)\}$ is a sequence in $f([a,b])$.

