I am given this problem:

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> Suppose that $\{C\_{\alpha}\}$ is a family of connected subspaces of a space $X$. If each intersection $C\_{\alpha} \cap C\_{\alpha'}$ is nonempty, then $\cup\_{\alpha} C\_{\alpha}$ is connected.
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And I am supposed to use the following lemma:

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> If $A \subseteq X$ is both open and closed, then any connected subspace $C \subseteq X$ that has a nonempty intersection with $A$ must be contained in $A$.
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Could any of you offer me a hint as how to proceed?