Closure of Connected Set is Connected

Question

I wanted To show that if E is connected subset then $\bar E$ is also Connected By Using Definition.
There is one answer regarding this question but that does not use Definition So this question is not repeated .
My attempt:
On contrary $\bar E$ is disconnected so $\bar E$=$A \cup B$ Where A and B are disjoint open set .I had only info about E is connected that means E can not be written like above .
From Hint:
E$\subset$ $\bar E$=$A \cup B$ So E=$(A\cap E) \cup (B\cap E)$
$(A\cap E)$ and $(B\cap E)$ are both open and disjoint Because In case
x$\in(A\cap E) \cap (B\cap E)$ Then x$\in A \cap B$ Which contradicts with Given .Is this is enough to show ?