I was wondering if it is true that every infinite totally disconnected compact Hausdorff space $X$ contains a convergent sequence $(x_n)_{n\ge 1}$ with limit point $x\in X$, that is, for any clopen neighborhood $O_x$ of $x$ there is an $N\ge 1$ such that $x_n\in O_x$ for all $n\ge N$.
Thank you very much for your input!