Consider the sequence defined by

$a_1=2,a_{n+1}=\frac{1}{3-a_n}$

Which of the following statements is correct?

$($a$)a_n$ is decreasing and bounded above by $\frac{1}{3}$

$($b$)a_n$ is increasing and bounded above by $5$

$($c$)a_n$ is decreasing and bounded below by $\frac{1}{2}$

$($d$)a_n$ is increasing and bounded below by $2$

$($e$)a_n$ is decreasing and bounded above by $2$