# Why does a null state correspond to a field that any correlator containing it vanishes?

I am reading the 7th chapter of Di Francesco's CFT book. It builds, for example in section 7.3, a null state |x> which is orthogonal to the whole Verma Module. The author asserts that the field x representing the state has the property that any correlation function containing the field x vanishes, which the author does not provide the reader with a justification.

Now I am confused by the assertion. Why is the assertion true? Can anyone give me an explanation? Thanks.

The two-point functions of the null state $N(z,\bar z)$ with an operator $O(z,\bar z)$ is proportional to the inner product of the corresponding states $\langle n| o\rangle$, so it vanishes. The three-point and higher-point functions $NOP\cdots$ vanish as well because the products $OP\cdots$ may be expanded, using OPEs, in terms of "single operators", and that is how the correlator is reduced to the two-point function.