Corollary

Let $(A_n)_{n \geq 0}$ be an infinite sequence of connected sets such that $A_{n+1} \cap A_n \neq \emptyset$ for all $n \geq 0$. Then the union $\bigcup_{n=0}^\infty A_n$ is connected.

By induction on $n$ we see immediately that the set $B_n = \bigcup_{i=0}^n A_i$ is connected for all $n$, by Proposition 2. The sets $B_n$ have a non-empty intersection; hence their union, equal to $\bigcup_{n=0}^\infty A_n$, is connected by Proposition 2.