Let $X$ be a topological space. Suppose that $U_{1}, U_{2}, \ldots$ are connected subsets of $X$ with $U_{j} \cap U_{1}$ non-empty for all $j>0$. Prove that

$$W=\bigcup_{j>0} U_{j}$$

is connected. If each $U_{j}$ is path-connected, prove that $W$ is path-connected.