Let $A_1, A_2, \dots $ connected subsets of $X$ s.t $A_n \cap A_{n+1} \ne \emptyset$ for all $n$. Show that the union of the sets $A_n$ is connected. 
Let $A_1, A_2, \dots $ be a sequence of connected subsets of a metric space $X$ such that $A_n \cap A_{n+1} \ne \emptyset$ for all $n$. Show that the union of the sets $A_n$ is connected.

So we want to show $\bigcup_\alpha ^n A_\alpha$ is connected. Assume the opposite so that $\bigcup_\alpha ^n A_\alpha$ is not connected, then $\bigcup_\alpha ^n A_\alpha = E \cup F$ for some sets $E$ and $F$. Pick $x \in \bigcup_\alpha ^n A_\alpha \implies x \in E$ or $x \in F$.
This is where I'm stuck, how can I proceed here?

Hint:

*

*Show that if $A, B$ are connected and $A \cap B \ne \emptyset$, then $A \cup B$ is connected.


*Use induction to prove your theorem.
Concerning 1: Pick $x \in A \cap B$. Then the connected component $C(x)$ of $x$ in the subspace $A \cup B$ contains the connected sets $A$ and $B$, thus $A \cup B \subset C(x) \subset A \cup B$. i.e. $C(x) = A \cup B$.
