Is $[a,b] \subset B$ necessarily true? I am currently working on a problem, and in order to continue, I need to first make sure if the following statement is true:
Suppose I have a set $A$ which is dense in the interval $[a,b]$. Now let $B$ be a closed set with $A \subseteq B$. Is it necessarily true that $[a,b]\subseteq B$?
My thinking is yes, but I’m struggling to convince myself why it is true. Can anyone please give me an indication if this is true and why?

Yes it's true. Let $x$ be any point in $[0,1]$. Then $x$ is a limit point of $A$ since $A$ is dense in $[0,1]$. Therefore $x$ is a limit point of $B$ since $A\subset B$. Since $B$ is closed it contains all of its limit points, therefore $x\in B$.
