if $\{a_n\}$ is a sequence such that $a_n \in [c,b]$ and $a_n \rightarrow a$ then $a \in [c,b]$
I just wanted to make sure I was writing a correct proof and communicating my ideas correctly.
Proof
Suppose $a \notin [c,b]$. Given that $a_n \rightarrow a$ this means: $$\forall \ \epsilon >0,\ \exists \ N\in \mathbb{N} \ s.t.\ \forall \ n \geq N \ |a_n - a| < \epsilon$$
$$\Rightarrow \ \exists \ a_n \ s.t. \ a_n \notin [c,b]$$
Not sure if I have to give this explanation after, but since $a_n$ converges to $a$ it means eventually that $a_n$ and $a$ are going to be so close or equal that $a_n$ will not be able to be in $[c,b]$