A space $X$ is step connected if given any open covering $\mathcal{U}$ of $X$ and any pair of points $p,q\in X$ there is a finite sequence $U_1,\ldots,U_n$ of sets belonging to $\mathcal{U}$ so that $p\in U_1$, $q\in U_n$ and $U_i\cap U_{i+1}\neq\emptyset$ for $1\leq j\leq n-1$.
Prove that a space is step connected if and only if it is connected.
So far I was able to prove that step connected implies connected by showing the negation, that disconnected implies not step connected. Which was pretty straight forward by taking the open cover to be the partition of $X$ into open sets. The other implication I am struggling with. My first thought was to fix a $q\in X$ and let $A$ be the set of points such that this step sequence can be found. Then showing $A$ is open, closed, and nonempty thus by $X$ being connected we would see that $A$ is all of $X$ and thus $X$ is step connected. I was able to show that $A$ is open and nonempty, but cannot seem to show that $A$ is closed. Any tips?
Also this isn't for a homework assignment, I am trying to prepare for an upcoming qualifying exam.