First, the only clopen sets are $X$ and $\phi$, because if a proper subset $A$ is clopen, then $A$ and $X-A$ are open, so X-A and A are closed and hence finite, giving as $X=A\cup (X-A)$ is finite, which is a contradiction.
Now, as X is infinite, there must be an infinite set $A$ such that its complement $X-A$ is also infinite. So $A$ is neither open nor closed. [A subset is open if its complement is finite, and closed if it is finite. $A$ is neither.]
Is this proof correct? I have intuitively guess the existence of such an $A$ and have not been able to actually prove it.