The function $f : (0, 1] \to\mathbb R$ is a bounded and continuous on $(0, 1]$. Let $\{x_n\}$ be a sequence in $[0, 1]$. Prove or disprove the following.
(a) If $\{x_n\}$ is convergent, then $\{f(x_n)\}$ is convergent.
(b) If $\{f(x_n)\}$ is convergent, then $\{x_n\}$ is convergent.
(c) $\{x_n\}$ has a subsequence $\{x_{n_k}\}$ such that $\{f(x_{n_k})\}$ is convergent.
I'm struggling with this question for a while now. And my first insight was that if $\{x_n\}$ is within (0, 1] then $\{x_n\}$ converging would imply $\{f(x_n)\}$ also converging as f is continuous. leading me to realize that if $\{x_n\}$ is a constant sequence of 0 s then $\{f(x_n)\}$ is undefined. I do believe that implies we can't find a limit for it. I want to know if my reasoning is correct first of all also I'm not sure how to rigorously prove it. if anyone can help it would be a great favor.