Let $X$ be a topological space. Prove that if $X=\bigcup\limits_{k=1}^{\infty}X_k$, $X_k$ is connected subset of $X$ and $X_k\cap X_{k+1}\neq\varnothing\quad\forall k\geq 1$, then $X$ is connected.
I can prove that if every pair $x,y$ of $X$ belongs to a connected subset of $A_{xy}$ then $X$ is connected.
I also proved if $\{A_{\alpha}\}_{\alpha\in I}$ is a family of connected subsets of $X$ and $\bigcap\limits_{\alpha\in I} A_{\alpha}\neq\varnothing$ then $\bigcup\limits_{\alpha\in I} A_{\alpha}$ is connected.
I found this link, but I don't understand at all. Could someone help me to shred light on this problem? Thanks in advance