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...



Since $[a,b]$ is compact, there exists a subsequence $\{ x_{n_k} \}$ converging to some $X \in [a,b]$.
Since $f$ is continuous on $[a,b]$, the subsequence $\{ f(x_{n_k}) \}$ converges to $f(X)$.
